JP3858262B2 - Rolling bearing rotation accuracy evaluation method and rolling bearing rotation accuracy evaluation apparatus - Google Patents

Description

ω_ｃ：ボールの公転周波数
【００２８】
式（１５）は少々の計算を経てＺω_ｃの整数倍の周波数成分のみで表される式（１６）に変換できる。
【数２】

ｍ＝１，２，３・・（次数成分）
ただし、Ａ_ｎ＝ＺＣ_ｎ／２と置き換えた。すなわち、ボール数Ｚの整数倍±１のうねりの山数の成分のみが振動となって現れることとなる。同様に変位プローブ２１より角度α（図１では９０度）だけ離れた方位に、変位プロ―ブ２２が設置されているとするとき、それで観測される弾性変位の時系列ｆ_α（ｔ）は、式（１７）のようになる。
【数３】

【００２９】
観測される弾性変位の時系列ｆ（ｔ）およびｆ_α（ｔ）の次数ｍのフーリエ展開係数をそれぞれＦ（ｍ）およびＦ_α（ｍ）とおけば、式（１６）および式（１７）より次数ｍの成分の係数との間に、式（１８）および式（１９）の関係が成り立つ。
Ａ_ｍＺ−１ｅ^−ｊθ＋Ａ_ｍＺ＋１ｅ^ｊθ＝Ｆ（ｍ） （１８）
Ａ_ｍＺ−１ｅ^{−ｊ（θ＋α）}＋Ａ_ｍＺ＋１ｅ^{ｊ（θ＋α）}＝Ｆ_α（ｍ） （１９）
【００３０】
ここで式（１８）と（１９）を、未知数Ａ_ｍＺ−１ｅ^−ｊθとＡ_ｍＺ＋１ｅ^ｊθの複素２元連立方程式とみなして解けば、以下の解が得られる。
【数４】

ただし、α≠ｎπ（ｎ：整数）でなければならない。以上を要約すれば、互いに１８０°でない角度をなす二つの方位の振動を測定して得られる時系列の各フーリエ展開係数を用いて、式（２０）に示す解にしたがって、固定輪の軌道のｍＺ−１山に起因する成分およびｍＺ＋１山に起因する成分の係数をそれぞれ求めることができ、さらにＲＭＳ値を次のように算出することができる。
ｍＺ−１山成分のＲＭＳ値＝√２│Ａ_ｍＺ−１│＝√２│Ａ_ｍＺ−１ｅ^−ｊθ│（21)
ｍＺ＋１山成分のＲＭＳ値＝√２│Ａ_ｍＺ＋１│＝√２│Ａ_ｍＺ＋１ｅ^ｊθ│ （22)
【００３１】
（実施例２）
次に、角速度ｍＺω_ｃの振動の最大振幅値と最大方位を求める。式（１６）を参照して、次数ｍと−ｍの成分和の振動を式（２３）のように表す。
ｆ_ｍ（ｔ）＝（Ａ_ｍＺ−１ｅ^−ｊθ＋Ａ_ｍＺ＋１ｅ^ｊθ）ｅ^ｊｍＺω _ｃ ^ｔ＋（Ａ_{−（ｍＺ−１）}ｅ^ｊθ＋Ａ_{−（ｍＺ＋１）}ｅ^−ｊθ）ｅ^{−ｊｍＺω} _ｃ ^ｔ (２３)
【００３２】
ここで、θを式（２３）が最大方位、すなわち最大振幅値を呈する前記固定輪軌道上のある基準点を仮定したときの方位角であるとする。Ａ_n（ｎ＝−∞〜∞）は実数関数のフーリエ展開係数であるからＡ_-n＝Ａ_n ^*（Ａ_nの複素共役）が成り立ち、また、絶対値と位相によりＡ_mZ-1＝｜Ａ_mZ-1｜ｅ^{j ψ}，Ａ_mZ+1＝｜Ａ_mZ+1｜ｅ^{j φ}
と表せるから、式（２３）は、式（２４）のように変形できる。
ｆ_m（ｔ）＝２｛｜Ａ_mZ-l｜ｃｏｓ（ｍＺω_cｔ−θ＋ψ）＋｜Ａ_mZ+1｜ｃｏｓ（ｍＺω_cｔ＋θ＋φ）｝ （２４）
【００３３】
式（２４）は、θ、ψ、φ間の条件により式（２５）のような最大振幅あるいは最小振幅の振動となる。
【数５】

ただし、ｎは整数である。
【００３４】
以上より、角速度ｍＺω_ｃの振動の最大ＲＭＳ値が√２（│Ａ_ｍＺ−１│＋│Ａ_ｍＺ＋１│）であり、最小ＲＭＳ値が√２││Ａ_ｍＺ−１│−│Ａ_ｍＺ＋１││となることがわかる。
【００３５】
振動の最大の方位を求めるには、２方位の次数ｍの係数の式（１８）と（１９）を用いる。変位プローブ２１より、角度γだけ離れた方位の振動の次数ｍの係数Ｆγ（ｍ）は式（２６）のように表せる。
Ｆ_γ（ｍ）＝｛Ｆ（ｍ）ｓｉｎ（α−γ）＋Ｆ_α（ｍ）ｓｉｎγ｝／ｓｉｎα（２６)
【００３６】
ここで振幅の２乗を求め、微分して０とおいて、これを満たすγを算出する。式（２６）より
【数４】

【００３７】
これより次式を得る。
【数５】

γ＝γ₀のとき｜Ｆ_γ（ｍ）│²は、式（２９）のように最大（復号が＋）または最小（復号が−）になる。
【数６】

【００３８】
式（２５）の説明で与えた最大ＲＭＳ値√２（｜Ａ_ｍＺ−１｜＋｜Ａ_ｍＺ＋１｜）と、最小ＲＭＳ値√２｜｜Ａ_ｍＺ−１｜＋｜Ａ_ｍＺ＋１｜｜は、√２｜Ｆ_γ（ｍ）│等しい。ただし、最大値を呈する方位および最小値を呈する方位は以下の方法による。
【００３９】
［方位の決定方法］
｜Ｆ（ｍ）│^２ｃｏｓ２α＋│Ｆ_α（ｍ）｜^２−｛Ｆ（ｍ）Ｆ^＊ _α（ｍ）＋Ｆ^＊（ｍ）Ｆ_α（ｍ）｝ｃｏｓα≦０のとき｜γ_０｜≦π／４ならば方位γ_０およびγ_０＋πで最大、γ_０±π／２で最小となり、π／４＜｜γ_０｜≦π／２ならば方位γ_０およびγ_０＋πで最小、γ_０±π／２で最大となる。また、｜Ｆ（ｍ）｜^２ｃｏｓ２α＋｜Ｆ_α（ｍ）｜^２−｛Ｆ（ｍ）Ｆ^＊ _α（ｍ）＋Ｆ^＊（ｍ）Ｆ_α（ｍ）｝ｃｏｓα＞０のとき｜γ_０｜≦π／４ならば方位γ_０およびγ_０＋πで最小、γ_０±π／２で最大となり、π／４＜｜γ_０｜≦π／２ならば方位γ_０およびγ_０＋πで最大、γ_０±π／２で最小となる。
【００４０】
例えば、変位プローブ２１と２２の間の角度が９０°、すなわちα＝π／２の場合を考えれば直感的に理解できる。最初の条件は｜Ｆ（ｍ）｜^２≧｜Ｆ_α（ｍ）｜^２のことである。｜γ_０｜≦π／４ならば変位プローブ２１から角度γ_０およびγ_０＋πが最大方位であり、π／４＜｜γ_０｜≦π／２ならば変位プローブ２１から角度γ_０±π／２が最大方位である。そして、最大方位と直交する方向が最小方位である。また、もう一方の条件は｜Ｆ（ｍ）｜^２＜｜Ｆ_α（ｍ）｜^２のことである。｜γ_０｜≦π／４ならば変位プローブ２１から角度γ_０±π／２が最大方位であり、π／４＜｜γ_０｜≦π／２ならば変位プローブ２１から角度γ_０およびγ_０＋πが最大方位である。そして、同じく最大方位と直交する方向が最小方位である。
【００４１】
このようにして決定された軸受の固定輪の最大方位または最小方位の位置に印をつける。印のつけ方については本発明には直接関係しないが、ケガキ、ペイント、打刻などが考えられる。
【００４２】
（実施例３）
回転輪の振動は、前記振動測定用センサーである測定プローブのひとつによって次式のように得られる。
【数９】

【００４３】
ただし、回転角速度をω_γとし、転動体の公転角速度をω_ｃとするとき、ω_ｉ＝ω_γ−ω_ｃで表せる。式（３０）は、少々の計算を経てボール数の整数倍±１の山数成分のみからなる式（３１）に変換できる。ただし、Ｂ_ｎ＝ＺＣ_ｎ／２と置く。
【数１０】

【００４４】
フーリエ変換から得られるｍＺ−１次とｍＺ＋１次のフーリエ展開係数Ｂ_ｍＺ−１ｅ^−ｊθおよびＢ_ｍＺ＋１ｅ^ｊθの絶対値│Ｂ_ｍＺ−１│および｜Ｂ_ｍＺ＋１｜は容易に計算できる。このときの回転輪と固定輪が逆に使用される時、ｍＺ−１次とｍＺ＋１次の成分は前記のようにｍＺω_ｃなるひとつの周波数成分に合成され、しかもその大きさは方位に依存する。この合成された振動の振幅の最大ＲＭＳ値および最小ＲＭＳ値はそれぞれ√２（│Ｂ_ｍＺ−１｜＋｜Ｂ_ｍＺ＋１｜）および√２││Ｂ_ｍＺ−１｜＋｜Ｂ_ｍＺ＋１｜｜となる。
【００４５】
（実施例４）
実施例１、２、３では、転がり軸受は回転するとき固定輪と回転輪の軌道が転動体との間で弾性変形を生じ、軌道のうねりがあるとき転がり軸受全体を剛体運動的に振動すると考えられる場合を説明した。転がり軸受単品での回転時には、この剛体運動的振動に加えて、転動体の通過に伴う外輪の外径面や内輪の内径面に微少の弾性変形による変位が観測される。特に、転動体通過の角周波数は前記Ｚω_ｃに一致しており、従来の測定ではこの複合的振動から前記固定輪の軌道うねりによる振動成分を区別して評価できない。
【００４６】
本実施例では、前記実施例と同様の測定結果Ｆ’（ｍ）とＦ’_α（ｍ）に加えて、図１で点線で示すような第３の変位プローブ２３を、第１の変位プローブ２１に対して角度βをなす位置に設置して、測定値ｆ_β（ｔ）の次数ｍのフーリエ展開係数Ｆ’_β（ｍ）を得る。このとき式（１８）、（１９）に対応してＡ_ｍＺ−１ｅ^−ｊθ、Ａ_ｍＺ＋１ｅ^ｊθ、および弾性変形成分に関するＤ_ｍＺｅ^{−ｊｍＺθ}を未知数とする次の複素３元連立方程式が成り立つ。
Ａ_ｍＺ−１ｅ^−ｊθ＋Ａ_ｍＺ＋１ｅ^ｊθ＋Ｄ_ｍＺｅ^{−ｊｍＺθ}＝Ｆ’（ｍ）
（３２）
Ａ_ｍＺ−１ｅ^{−ｊ（θ＋α）}＋Ａ_ｍＺ＋１ｅ^{ｊ（θ＋α）}＋Ｄ_ｍＺｅ^{−ｊｍＺ（θ＋α）}＝Ｆ’_α（ｍ）（３３）
Ａ_ｍＺ−１ｅ^{−ｊ（θ＋β）}＋Ａ_ｍＺ＋１ｅ^{ｊ（θ＋β）}＋Ｄ_ｍＺｅ^{−ｊｍＺ（θ＋β）}＝Ｆ’_β（ｍ）（３４）
Δ＝ｅ^{ｊ（α−ｍＺβ）}＋ｅ^{−ｊ（β＋ｍＺα）}−ｅ^{−ｊ（α＋ｍＺβ）}−ｅ^{ｊ（β−ｍＺα）}＋ｅ^{−ｊ（α−β）}−ｅ^{ｊ（α−β）}≠０と仮定すると、解から次のように各成分の振幅を計算することができる。
【００４７】
│Ａ_ｍＺ−１│＝│Ａ_ｍＺ−１ｅ^−ｊθ│＝│｛（ｅ^{ｊ（α−ｍＺβ）}−ｅ^{ｊ（β−ｍＺα）}）Ｆ’（ｍ）＋（ｅ^ｊβ−ｅ^{−ｊｍＺβ}）Ｆ’_α（ｍ）＋（ｅ^{−ｊｍＺα}−ｅ^ｊα）Ｆ’_β（ｍ）｝／Δ│ （３５）
│Ａ_ｍＺ＋１│＝│Ａ_ｍＺ＋１ｅ^−ｊθ│＝│｛（ｅ^{−ｊ（β＋ｍＺα）}−ｅ^{−ｊ（α＋ｍＺβ）}）Ｆ’（ｍ）＋（ｅ^{−ｊｍＺβ}−ｅ^−ｊβ）Ｆ’_α（ｍ）＋（ｅ^−ｊα−ｅ^{−ｊｍＺα}）Ｆ’_β（ｍ）｝／Δ│ （３６）
したがって、前記実施例と同様に最大ＲＭＳ値は√２（│Ａ_ｍＺ−１｜＋｜Ａ_ｍＺ＋１｜）、最小ＲＭＳ値は√２││Ａ_ｍＺ−１｜−｜Ａ_ｍＺ＋１｜｜として求められる。
【００４８】
次に最大（最小）の振動の方位を求める方法を説明する。複素３元連立方程式の第３の解は次の式（３７）のように得られる。
Ｄ_mZｅ^{-jmZ θ}＝｛（ｅ^{-j( α - β )}−ｅ^{j( α - β )} ）Ｆ’（ｍ）＋（ｅ^{-j β}−ｅ^{j β}）Ｆ’_α（ｍ）＋（ｅ^{j α}−ｅ^{-j α}）Ｆ’_β（ｍ）｝／Δ （３７）
【００４９】
前記測定結果Ｆ’（ｍ）とＦ’_α＝（ｍ）および式（３７）の結果を用いて、式（３８）と（３９）
Ｆ（ｍ）＝Ｆ’（ｍ）−Ｄ_ｍＺｅ^{−ｊｍＺθ} （３８）
Ｆ_α（ｍ）＝Ｆ’_α（ｍ）−Ｄ_ｍＺｅ^{−ｊｍＺ（θ＋α）} （３９）
を計算し、式（３８）と（３９）の結果に対して実施例２の諸式と方法を用いれば、固定輪の角周波数ｍＺω_ｃ成分の最大（または最小）方位を求めることができる。
【００５０】
以上の本実施例の説明では触れていないが、二つの異なる方位は必ずしも直角である必要はない。また、異なる二方位の振動を測定できるひとつのセンサーを用いることもできる。さらにはセンサは変位プローブに限らず、微分・積分関係にある他のセンサを用いてもよいし、変位のみならず速度や加速度での評価もありうる。振動の信号の微分あるいは積分とフーリエ展開係数に対する計算の関係は当該関係者には明らかであるから、これらに関する説明は省略する。
【００５１】
【発明の効果】
本発明によれば、定速度で回転する転がり軸受の固定輪の二つの異なる方位のラジアル方向振動のフーリエ展開係数を用いて、固定輪軌道形状に起因する振動成分の振幅の最大（または最小）値を計算して評価でき、その方位を算定して固定輪上に印をつけた転がり軸受を提供できる。
【００５２】
また、本発明により、転がり軸受の評価時の回転輪が固定輪として使用される場合の固定輪軌道形状に起因する振動成分の振幅の最大（または最小）値を計算する方法を提供できる。これにより、本発明に基づく転がり軸受の評価を実施する装置は、外輪または内輪の一方のみ回転できればよく、安価に実現できる。
【００５３】
さらに、転動体の通過に伴う固定輪の弾性変形振動の重畳が無視できない測定状況に対して、前記二つの異なる方位のラジアル方向振動のフーリエ展開係数と第３の方位のラジアル方向振動のフーリエ展開係数を用いることにより、固定輪軌道形状に起因する振動成分の振幅の最大（または最小）値を前記弾性変形振動の影響を除くように計算して評価でき、その方位を算定して固定輪上に印をつけた転がり軸受を提供できる。
【図面の簡単な説明】
【図１】本実施の形態にかかる試験装置のブロック図である。
【図２】試験装置の正面図である。
【符号の説明】
１０ 軸受
２１，２２，２３ 変位プローブ
３１，３２，３２ 変位測定装置
４１ コンピュータ
５１ モータ
５２ カップリング５２
５３ スピンドル
５４ 回転軸[0001]
BACKGROUND OF THE INVENTION
The present invention relates to a technique for evaluating radial vibrations asynchronous to the rotation of a rotating body such as a bearing and a spindle incorporating the bearing based on frequency analysis, and in particular, the amplitude depends on the radial direction. The present invention relates to a technique for evaluating a maximum (minimum) amplitude value and a maximum (minimum) direction of a specific frequency component that changes.
[0002]
[Prior art]
Rolling bearings or rotating bodies such as spindles incorporating them generate vibrations due to roundness shape errors and the like, which can be a significant vibration source for structures such as machine tools that use them as components. is there. On the other hand, in precision parts such as hard disk devices, the radial vibration of the ball bearing used for the rotating shaft of the disk mainly causes positioning errors of the magnetic head. It has been demanded.
[0003]
By the way, the vibration component asynchronous to the rotation among the vibrations generated when the rotating body rotates at a constant speed is said to be a rotation asynchronous (NRRO: Non Repeatable Run Out) vibration component. In order to manufacture and supply ball bearings or spindles for machine tools with excellent rotational accuracy for hard disk drives, it is necessary to quantitatively evaluate the radial NRRO vibration component and eliminate products outside the allowable range. There is. Furthermore, it is also necessary to selectively evaluate an NRRO vibration component having a specific frequency in the vicinity of the resonance frequency of the hard disk device.
[0004]
Here, the radial NRRO vibration component of a rolling bearing or a rotating body such as a spindle incorporating these is unique in terms of geometrical dimensions and shape accuracy of parts such as outer and inner rings and rolling elements of the rolling bearing and a constant speed rotation frequency. It consists of vibrations of various frequencies determined by When observing the radial vibration of a rolling bearing at one point on the fixed ring, the vibration caused by the shape error of the rotating wheel and rolling element is equally observed at any position on the fixed ring. It is known that the vibration caused by the error is observed with a different magnitude depending on the measurement position. Such vibration cannot be evaluated correctly unless the maximum amplitude is found over the entire circumference of the fixed ring. To this end, it is possible to find a maximum value of a specific frequency component by repeating frequency analysis while relatively displacing the vibration measurement sensor and the bearing or spindle in the circumferential direction. Not only that, but it takes time to measure.
[0005]
Next, for example, when the inner ring rotates, if the outer ring orbit, which is a fixed ring, has a slight undulation, vibration of one specific frequency is generated by a pair of undulations. A pair of vibrations corresponding to the number of peaks is generated. At the same time, vibration of one specific frequency is generated due to the pair of undulations of the track of the inner ring which is a fixed ring. That is, depending on how the rolling bearing is used, it may be necessary to strictly select one or both of a specific frequency vibration and a pair of frequency vibrations as harmful vibration components. For this purpose, providing a mechanism capable of observing both the inner ring rotation and the outer ring rotation leads to the complexity and cost of the test apparatus.
[0006]
One of the objects of the present invention is to provide a method for obtaining each vibration value corresponding to each mountain number from the vibration of one specific frequency caused by the pair of mountain waves of the undulation of the track of the fixed ring. is there. In order to achieve this purpose, the vibration of the bearing is observed simultaneously from two radial directions that are not opposite (that is, not 180 °), and the vibration value at a specific frequency caused by the fixed ring obtained based on the frequency analysis result is a coefficient. Gives a method to calculate from the solution of simultaneous equations.
[0007]
Another object of the present invention is to provide a maximum (minimum) value of vibration at a specific frequency caused by a pair of peaks of a fixed ring based on a pair of vibration values of the fixed ring obtained as described above, and a maximum value representing the maximum value. It is to provide a method for calculating the (minimum) orientation.
[0008]
On the contrary, another object of the present invention is based on the measurement of a vibration component of a pair of frequencies based on a pair of peaks exhibited by a rotating wheel. It is to provide a way to estimate and evaluate (minimum) values.
[0009]
Furthermore, since the fixed ring is not a rigid body, the measurement point on the circumference thereof observes vibration accompanied by the elastic deformation of the fixed ring due to the passage of the rolling element. Since it is equal to a specific frequency due to the shape error, there is a fear that the vibration component due to the shape error of the fixed ring cannot be accurately evaluated. Even in such a situation, it is possible to provide a method for accurately evaluating the vibration component of the fixed ring by removing the influence of the passing component of the rolling element by adding observation of vibration in a different third direction. Is another purpose.
[0010]
In this way, a bearing is provided in which the vibration is marked at the maximum position or at the minimum position 90 ° away from it so that the spindle or motor with the least vibration in a specific direction can be manufactured. This is yet another object of the present invention.
[0011]
[Means for Solving the Problems]
  The method for evaluating the rotational accuracy of a rolling bearing according to the present invention is obtained by measuring the radial vibration of a fixed ring of a rotating rolling bearing using two vibration measuring sensors installed at a phase α in the circumferential direction. The obtained sensor signal is discretized via an A / D converter to obtain two synchronized digital data, the digital data is Fourier transformed, and the angular velocity Zω obtained by Fourier transformation is obtained._cVibration value F (m), F of order m_αUsing (m), based on
  A_mZ-1e^{-j θ}= {E^{j α}F (m) -F_α(M)} / 2jsin α (1)
  A_{mZ + 1}e^{j θ}= {F_α(M) -e^{-j α}F (m)} / 2jsinα (2)
(Where m is the vibration order, Z is the number of rolling elements, ω_cIs the angular velocity of revolution of the rolling element, θ is the central angle between the unknown reference position on the fixed ring and one of the vibration measuring sensors,F (m) is a vibration value of order m of one of the vibration measurement sensors, F _α (M) is the vibration value of the order m of the other vibration measurement sensor arranged at the phase α with respect to the one vibration measurement sensor.)
Unknown A_mZ-1e^{-j θ}, A_{mZ + 1}e^{j θ}From this unknown, the RMS value of the vibration component due to mZ-1 peak and mZ + 1 peak is obtained from the following equation,
  RMS value of mZ-1 peak component = √2 | A_mZ-1| = √2 | A_mZ-1e^{-j θ}│ (3)
  RMS value of mZ + 1 peak component = √2 | A_{mZ + 1}| = √2 | A_{mZ + 1}e^{j θ}│ (4)
The rotational accuracy is evaluated from these RMS values.
[0012]
[Action]
Computers as two sample value series synchronized with two vibration measurement sensors arranged via two A / D converters to observe radial vibrations at two points on the fixed ring of a rolling bearing The solution of the complex binary linear equation that takes the complex function value of the desired frequency component selected from each Fourier transform and the function of the angle between the positions of the sensors for vibration measurement as the fixed ring shape It is set as the vibration value (RMS value) corresponding to the number of pairs of peaks. The sum (difference) of the absolute values of these two vibration values is set as the maximum (minimum) value of vibration depending on the shape of the fixed ring, and the complex value of the desired frequency component selected from the respective Fourier transform and vibration By calculating the maximum (minimum) direction of vibration as an angle relative to the vibration measurement sensor from the angle between the measurement sensor arrangement positions, the maximum vibration value and its direction can be determined, and compared with the threshold value. Thus, the rotational accuracy of the rolling bearing can be evaluated.
[0013]
Angular velocity mZω_cMaximum amplitude value and minimum amplitude value of
Maximum RMS value = √2 (| A_mZ-1| + | A_{mZ + 1}|) (5)
Minimum RMS value = √2 || A_mZ-1|-| A_{mZ + 1}|| (6)
Can be expressed as
[0014]
  Furthermore, angular velocity mZω_cThe phase of the maximum amplitude value and the minimum amplitude value of
  | F (m) |²cos2α + | F_α(M) │²-{F (m) F^* _α(M) + F^*(M) F_α(M)} When cos α ≦ 0, | γ₀If ≦≦ π / 4, γ₀And γ₀+ Π is maximum and γ₀± π / 2 is minimum, and π / 4 <| γ₀If ≦≦ π / 2, γ₀And γ₀+ Π and γ₀Maximum at ± π / 2
,
  | F (m) |²cos2α + | F_α(M) │²-{F (m) F^* _α(M) + F^*(M) F_α(M)} cos α> 0, | γ₀If ≦≦ π / 4, γ₀And γ₀+ Π minimum and γ₀± π / 2 is the maximum, and π / 4 <| γ₀If ≦≦ π / 2, γ₀And γ₀+ Π maximum, γ₀± π / 2 is minimum (however, 2γ₀Is given by equation (28)). It should be noted that F used in this specification (including claims)^*(M) is for F (m)ComplexBe conjugate,F ^* _α (M) is F _α The complex conjugate of (m)To do.
[0015]
Further, using the vibration value of the rotating wheel, the RMS value of the vibration component caused by the mZ-1 peak and the mZ + 1 peak when the rotating wheel and the fixed wheel are used oppositely according to the equation (31) is obtained.
RMS value of mZ-1 peak component = √2 | B_mZ-1| (7)
RMS value of mZ + 1 peak component = √2 | B_{mZ + 1}| (8)
Angular velocity mZω_cMaximum amplitude value and minimum amplitude value of
Maximum RMS value = √2 (| B_mZ-1| + | B_{mZ + 1}|) (9)
Minimum RMS value = √2 || B_mZ-1|-| B_{mZ + 1}|| (10)
It is what.
[0016]
That is, the sum of the absolute values of a pair of vibration values corresponding to the number of crests of the rotating wheel selected from the at least one Fourier transform of the one specific frequency when it becomes a fixed wheel. Gives the maximum (minimum) value of vibration.
[0017]
Furthermore, another vibration sensor is provided at the phase β, and the RMS value of the vibration component caused by the mZ−1 peak and the mZ + 1 peak is calculated by the equations (32) to (36), respectively.
RMS value of mZ-1 peak component = √2 | A_mZ-1| (11)
RMS value of mZ + 1 peak component = √2 | A_{mZ + 1}| (12)
Angular velocity mZω_cMaximum amplitude value and minimum amplitude value of
Maximum RMS value = √2 (| A_mZ-1| + | A_{mZ + 1}|) (13)
Minimum RMS value = √2 || A_mZ-1|-| A_{mZ + 1}|| (14)
And
[0018]
That is, a third vibration measurement sensor is arranged, and the same as the signals from the two vibration measurement sensors and simultaneously taken into the computer as three sample value series synchronized via the A / D converter, The complex value of the desired frequency component selected from each Fourier transform and the solution of the complex ternary linear equation which is a function of the angle between the three sensors for vibration measurement are a pair of peaks in the shape of a fixed ring. The vibration value corresponding to the number and the elastic deformation of the fixed ring accompanying the passage of the rolling elements is used, and the fixed ring is similarly evaluated.
[0019]
Similarly, angular velocity mZω_cThe phase of the maximum amplitude value and the minimum amplitude value of
F (m) = F ′ (m) −D_mze^−jmZθAnd F_α(M) = F ′_α(M) -D_mze^{−jmZ (θ + α)}│F (m) │²cos2α + │F_α(M) │²-{F (m) F^* _α(M) + F^*(M) F_α(M)} When cos α ≦ 0, | γ₀If ≦≦ π / 4, γ₀And γ₀+ Π is maximum and γ₀± π / 2 is minimum, and π / 4 <| γ₀If ≦≦ π / 2, γ₀And γ₀+ Π minimum and γ₀± π / 2 is the maximum, | F (m) |²cos2α + │F_α(M) │²-{F (m) F^* _α(M) + F^*(M) F_α(M)} cos α> 0, | γ₀If ≦≦ π / 4, γ₀And γ₀+ Π minimum and γ₀± π / 2 is the maximum, and π / 4 <| γ₀If ≦≦ π / 2, γ₀And γ₀+ Π maximum, γ₀± π / 2 is minimum (however, 2γ₀Is given by equation (28)).
[0020]
Based on the evaluation by the method for evaluating rotational accuracy as described above, in a rolling bearing, when the rolling bearing is installed by marking on the fixed ring of the maximum position or the minimum position of the vibration component, the processing machine By adjusting the cutting direction of the grinding wheel and the minimum direction of vibration of the bearing, or by adjusting the head movement direction and the minimum direction of bearing vibration in hard disks, the influence of vibration caused by rolling bearings can be reduced. I can do it.
[0021]
DETAILED DESCRIPTION OF THE INVENTION
DESCRIPTION OF EMBODIMENTS Hereinafter, a rolling bearing rotational accuracy test apparatus according to an embodiment of the present invention will be described in detail with reference to the drawings. FIG. 1 is a block diagram of a test apparatus according to the present embodiment, and FIG. 2 is a front view of the test apparatus.
[0022]
In the figure, a test apparatus 100 includes two displacement probes 21 and 22 that are non-contact optical, two displacement measuring apparatuses 31 and 32, a computer 41, a motor 51, and a coupling connected to a motor rotation shaft. 52 and a spindle 53 that rotatably supports a rotating shaft 54 connected to the coupling 52.
[0023]
The rotating shaft 54 is supported horizontally with respect to the spindle 53, and the rolling bearing 10 is fitted to the end of the rotating shaft 54 protruding from the end face of the spindle 53. Is pressed. The two displacement probes 21 and 22 are respectively arranged on the outer periphery of the fixed ring (outer ring) 11 of the rolling bearing 10 so that the sensor axis is orthogonal to the rolling bearing axis and in the vertical direction (y direction in FIG. 1) and the horizontal direction (see FIG. 1 in the x direction).
[0024]
The measurement is performed by rotating the rotating shaft 54 at a constant speed via the coupling 52 by the motor 51. At this time, the displacement probe 22 detects the vibration component in the radial direction of the fixed ring 11 and in the vertical direction, and transmits the detected vibration component to the displacement measurement device 32 as an electrical signal. The displacement measurement device 32 receives the vibration component. The electrical signal is converted into a voltage signal corresponding to the amount of displacement of the fixed wheel 11 in the vertical direction and transmitted to the computer 41. On the other hand, the displacement probe 21 detects a vibration component in the radial direction of the fixed wheel 11 and in the horizontal direction, and transmits the detected vibration component to the displacement measurement device 31 as an electric signal. The signal is converted into a voltage signal corresponding to the horizontal displacement of the fixed wheel 11 and transmitted to the computer 41.
[0025]
The computer 41 converts the voltage signal into a synchronous digital value by the built-in A / D converter and stores it. Further, the computer 41 performs Fourier transform on the stored digital value, extracts a predetermined frequency component, calculates a desired vibration value, and performs the following evaluation.
[0026]
The evaluation method of the present invention described below is actually based on a discrete Fourier transform on the computer 41 from a sequence of digital values discretized by an A / D converter (not shown). However, for ease of understanding, description will be made based on the Fourier series expansion of a continuous analog periodic signal.
[0027]
Here, as the rotating ring (inner ring) 12 of the ball bearing 10 rotates, the Z balls 13 trace an angular velocity ω while tracing the track of the outer ring 11 which is a fixed ring._cSuppose you are orbiting (revolution). As the outer ring 11 swells around the track, periodic elastic displacement occurs between the balls and radial vibrations are generated. The time series Fourier expansion coefficient of this periodic elastic displacement when a certain reference point on the orbit is assumed is expressed as C_nAssuming that (n = −∞ to ∞) and the displacement probe 21 is installed in a direction away from the reference point by an angle θ, the time series f (t) of the observed elastic displacement (that is, the fixed ring 11) Frequency component) is expressed by the following equation.
[Expression 1]

ω_c: Revolution frequency of the ball
[0028]
Equation (15) goes through a little calculation and Zω_cCan be converted into Expression (16) represented only by a frequency component that is an integral multiple of.
[Expression 2]

m = 1, 2, 3, (order component)
However, A_n= ZC_nReplaced with / 2. That is, only the component of the number of undulations of the integral multiple ± 1 of the number of balls Z appears as vibration. Similarly, when the displacement probe 22 is installed in the direction away from the displacement probe 21 by an angle α (90 degrees in FIG. 1), the time series f of the elastic displacement observed therewith_α(T) is as shown in Expression (17).
[Equation 3]

[0029]
Time series f (t) and f of the observed elastic displacement_αThe Fourier expansion coefficients of order m of (t) are respectively expressed as F (m) and F_αIf (m) is entered, the relationship of the equations (18) and (19) is established between the coefficients of the components of the order m from the equations (16) and (17).
A_mZ-1e^−jθ+ A_{mZ + 1}e^jθ= F (m) (18)
A_mZ-1e^{−j (θ + α)}+ A_{mZ + 1}e^{j (θ + α)}= F_α(M) (19)
[0030]
Here, equations (18) and (19) are converted into the unknown A_mZ-1e^−jθAnd A_{mZ + 1}e^jθThe following solution can be obtained by solving as a complex binary simultaneous equation.
[Expression 4]

However, α ≠ nπ (n: integer) must be satisfied. In summary, using the time series Fourier expansion coefficients obtained by measuring vibrations in two directions that are not at an angle of 180 ° with each other, according to the solution shown in Equation (20), The coefficients of the component due to mZ-1 peak and the component due to mZ + 1 peak can be respectively obtained, and the RMS value can be calculated as follows.
RMS value of mZ-1 peak component = √2 | A_mZ-1│ = √2│A_mZ-1e^−jθ│ (21)
RMS value of mZ + 1 peak component = √2 | A_{mZ + 1}│ = √2│A_{mZ + 1}e^jθ│ (22)
[0031]
(Example 2)
Next, angular velocity mZω_cThe maximum amplitude value and maximum direction of vibration are obtained. With reference to Expression (16), the vibration of the sum of the components of the order m and −m is expressed as Expression (23).
f_m(T) = (A_mZ-1e^−jθ+ A_{mZ + 1}e^jθ) E^jmZω _c ^t+ (A_-(MZ-1)e^jθ+ A_{− (MZ + 1)}e^−jθ) E^-JmZω _c ^t                                        (23)
[0032]
  Here, θ is assumed to be the azimuth when Equation (23) assumes a maximum azimuth, that is, a certain reference point on the fixed ring track exhibiting the maximum amplitude value. A_nSince (n = −∞ to ∞) is a Fourier expansion coefficient of a real function, A_-n= A_n ^*(A_nofComplexConjugate), and the absolute value and the phase_mZ-1= | A_mZ-1｜ e^{j ψ}, A_{mZ + 1}= | A_{mZ + 1}｜ e^{j φ}
Therefore, Expression (23) can be transformed as Expression (24).
      f_m(T) = 2 {| A_mZ-l｜ cos (mZω_ct−θ + ψ) + | A_{mZ + 1}｜ cos (mZω_ct + θ + φ)} (24)
[0033]
Expression (24) becomes a vibration with the maximum amplitude or the minimum amplitude as shown in Expression (25) depending on the conditions among θ, ψ, and φ.
[Equation 5]

However, n is an integer.
[0034]
From the above, angular velocity mZω_cThe maximum RMS value of vibration is √2 (│A_mZ-1│ + │A_{mZ + 1}│) and minimum RMS value is √2││A_mZ-1│-│A_{mZ + 1}It can be seen that ││.
[0035]
In order to obtain the maximum direction of vibration, equations (18) and (19) of the coefficient of the degree m in two directions are used. The coefficient Fγ (m) of the order m of vibration in the direction away from the displacement probe 21 by the angle γ can be expressed as shown in Expression (26).
F_γ(M) = {F (m) sin (α−γ) + F_α(M) sin γ} / sin α (26)
[0036]
  Here, the square of the amplitude is obtained, differentiated to 0, and γ satisfying this is calculated. From equation (26)
[Expression 4]

[0037]
  From this, the following equation is obtained.
[Equation 5]

  γ = γ₀When | F_γ(M) │²Becomes maximum (decoding is +) or minimum (decoding is-) as shown in Equation (29).
[Formula 6]

[0038]
Maximum RMS value √2 (| A given in the description of equation (25)_mZ-1| + | A_{mZ + 1}|) And minimum RMS value √2 || A_mZ-1| + | A_{mZ + 1}|| is √2 | F_γ(M) | However, the orientation that exhibits the maximum value and the orientation that exhibits the minimum value are determined by the following method.
[0039]
[Direction determination method]
| F (m) |²cos2α + │F_α(M) |²-{F (m) F^* _α(M) + F^*(M) F_α(M)} cos α ≦ 0 | γ₀If ≦≦ π / 4, orientation γ₀And γ₀+ Π maximum, γ₀± π / 2 is minimum, and π / 4 <| γ₀If ≦≦ π / 2, orientation γ₀And γ₀+ Π minimum, γ₀Maximum at ± π / 2. Also, | F (m) |²cos2α + | F_α(M) |²-{F (m) F^* _α(M) + F^*(M) F_α(M)} cos α> 0 | γ₀If ≦≦ π / 4, orientation γ₀And γ₀+ Π minimum, γ₀± π / 2 is the maximum, and π / 4 <| γ₀If ≦≦ π / 2, orientation γ₀And γ₀+ Π maximum, γ₀± π / 2 is the minimum.
[0040]
For example, if the angle between the displacement probes 21 and 22 is 90 °, that is, α = π / 2, it can be understood intuitively. The first condition is | F (m) |²≧ | F_α(M) |²That is. ｜ γ₀If | ≦ π / 4, the angle γ from the displacement probe 21₀And γ₀+ Π is the maximum orientation, and π / 4 <| γ₀If | ≦ π / 2, the angle γ from the displacement probe 21₀± π / 2 is the maximum orientation. The direction orthogonal to the maximum azimuth is the minimum azimuth. The other condition is | F (m) |²<| F_α(M) |²That is. ｜ γ₀If | ≦ π / 4, the angle γ from the displacement probe 21₀± π / 2 is the maximum orientation, and π / 4 <| γ₀If | ≦ π / 2, the angle γ from the displacement probe 21₀And γ₀+ Π is the maximum orientation. Similarly, the direction orthogonal to the maximum direction is the minimum direction.
[0041]
The position of the maximum azimuth or minimum azimuth of the fixed ring of the bearing thus determined is marked. Although the method of marking is not directly related to the present invention, scribing, painting, or engraving can be considered.
[0042]
(Example 3)
The vibration of the rotating wheel is obtained by the following equation by one of the measurement probes which are the vibration measurement sensors.
[Equation 9]

[0043]
However, the rotational angular speed is ω_γAnd the revolution angular velocity of the rolling element is ω_cWhere ω_i= Ω_γ−ω_cIt can be expressed as Expression (30) can be converted into Expression (31) consisting of only the peak number component of an integer multiple ± 1 of the number of balls through a little calculation. However, B_n= ZC_nPut / 2.
[Expression 10]

[0044]
MZ-1 order and mZ + 1 order Fourier expansion coefficients B obtained from Fourier transform_mZ-1e^−jθAnd B_{mZ + 1}e^jθAbsolute value of │B_mZ-1│ and | B_{mZ + 1}| Can be easily calculated. When the rotating wheel and the fixed wheel at this time are used in reverse, the mZ-1 order and mZ + 1 order components are mZω as described above._cIs synthesized into one frequency component, and its size depends on the direction. The maximum RMS value and the minimum RMS value of the amplitude of the synthesized vibration are respectively √2 (| B_mZ-1| + | B_{mZ + 1}│) and √2││B_mZ-1| + | B_{mZ + 1}||
[0045]
(Example 4)
In Examples 1, 2, and 3, when the rolling bearing rotates, the raceway of the fixed ring and the rotating ring causes elastic deformation between the rolling elements, and when there is swell of the raceway, the entire rolling bearing vibrates in a rigid body motion. Explained the possible cases. During rotation of a single rolling bearing, in addition to this rigid body vibration, displacement due to minute elastic deformation is observed on the outer diameter surface of the outer ring and the inner diameter surface of the inner ring as the rolling element passes. In particular, the angular frequency of passing through the rolling element is Zω._cIn the conventional measurement, the vibration component due to the track waviness of the fixed ring cannot be distinguished and evaluated from this complex vibration.
[0046]
In the present embodiment, the measurement results F ′ (m) and F ′ similar to those in the previous embodiment._αIn addition to (m), a third displacement probe 23 as shown by a dotted line in FIG. 1 is installed at a position that forms an angle β with respect to the first displacement probe 21, and the measured value f_βFourier expansion coefficient F ′ of order m of (t)_β(M) is obtained. At this time, A corresponding to equations (18) and (19)_mZ-1e^−jθ, A_{mZ + 1}e^jθ, And D for elastic deformation components_mZe^−jmZθThe following complex ternary simultaneous equations are established, where is an unknown.
A_mZ-1e^−jθ+ A_{mZ + 1}e^jθ+ D_mZe^−jmZθ= F '(m)
(32)
A_mZ-1e^{−j (θ + α)}+ A_{mZ + 1}e^{j (θ + α)}+ D_mZe^{−jmZ (θ + α)}= F ’_α(M) (33)
A_mZ-1e^{−j (θ + β)}+ A_{mZ + 1}e^{j (θ + β)}+ D_mZe^{−jmZ (θ + β)}= F ’_β(M) (34)
Δ = e^{j (α-mZβ)}+ E^{−j (β + mZα)}-E^{−j (α + mZβ)}-E^{j (β-mZα)}+ E^{−j (α−β)}-E^{j (α-β)}Assuming ≠ 0, the amplitude of each component can be calculated from the solution as follows.
[0047]
│A_mZ-1│ = │A_mZ-1e^−jθ│ = │ {(e^{j (α-mZβ)}-E^{j (β-mZα)}) F ′ (m) + (e^jβ-E^−jmZβ) F ’_α(M) + (e^-JmZα-E^jα) F ’_β(M)} / Δ | (35)
│A_{mZ + 1}│ = │A_{mZ + 1}e^−jθ│ = │ {(e^{−j (β + mZα)}-E^{−j (α + mZβ)}) F ′ (m) + (e^−jmZβ-E^−jβ) F ’_α(M) + (e^−jα-E^-JmZα) F ’_β(M)} / Δ | (36)
Accordingly, the maximum RMS value is √2 (| A_mZ-1| + | A_{mZ + 1})), Minimum RMS value is √2││A_mZ-1|-| A_{mZ + 1}It is calculated | required as ||.
[0048]
  Next, a method for obtaining the maximum (minimum) vibration direction will be described. The third solution of the complex ternary simultaneous equations is obtained as the following equation (37).
  D_mZe^{-jmZ θ}= {(E^{-j ( α - β )}-E^{j ( α - β )} )F ′ (m) + (e^{-j β}-E^{j β}) F ’_α(M) + (e^{j α}-E^{-j α}) F ’_β(M)} / Δ (37)
[0049]
The measurement results F ′ (m) and F ′_α= (M) and the result of Equation (37) are used to obtain Equations (38) and (39)
F (m) = F ′ (m) −D_mZe^−jmZθ                            (38)
F_α(M) = F ′_α(M) -D_mZe^{−jmZ (θ + α)}                  (39)
And using the equations and methods of Example 2 for the results of equations (38) and (39), the angular frequency mZω of the fixed ring_cThe maximum (or minimum) orientation of the component can be determined.
[0050]
Although not mentioned in the above description of the present embodiment, the two different orientations do not necessarily have to be perpendicular. One sensor that can measure vibrations in two different directions can also be used. Furthermore, the sensor is not limited to the displacement probe, and other sensors having a differential / integral relationship may be used, and evaluation may be performed not only by displacement but also by speed and acceleration. Since the relationship between the differentiation or integration of the vibration signal and the calculation with respect to the Fourier expansion coefficient is obvious to the concerned parties, explanations thereof will be omitted.
[0051]
【The invention's effect】
According to the present invention, the maximum (or minimum) amplitude of the vibration component due to the fixed ring raceway shape is obtained using the Fourier expansion coefficient of the radial vibration in two different directions of the fixed ring of the rolling bearing rotating at a constant speed. It is possible to provide a rolling bearing in which values can be calculated and evaluated, and the bearings can be calculated and marked on a fixed ring.
[0052]
Further, according to the present invention, it is possible to provide a method for calculating the maximum (or minimum) value of the amplitude of the vibration component caused by the shape of the fixed ring raceway when the rotating wheel at the time of evaluation of the rolling bearing is used as a fixed ring. Thereby, the apparatus for performing the evaluation of the rolling bearing according to the present invention only needs to be able to rotate only one of the outer ring or the inner ring, and can be realized at low cost.
[0053]
Furthermore, for the measurement situation in which the superposition of elastic deformation vibration of the fixed ring accompanying the passing of the rolling element cannot be ignored, the Fourier expansion coefficient of the radial vibration in the two different directions and the Fourier expansion of the radial vibration in the third direction By using the coefficient, the maximum (or minimum) value of the amplitude of the vibration component due to the shape of the fixed ring track can be calculated and evaluated so as to exclude the influence of the elastic deformation vibration. Rolling bearings marked with can be provided.
[Brief description of the drawings]
FIG. 1 is a block diagram of a test apparatus according to an embodiment.
FIG. 2 is a front view of the test apparatus.
[Explanation of symbols]
10 Bearing
21, 22, 23 Displacement probe
31, 32, 32 Displacement measuring device
41 computer
51 motor
52 Coupling 52
53 spindle
54 Rotating shaft

Claims (7)

The radial vibration of the fixed ring of the rotating rolling bearing is measured by using two vibration measuring sensors installed in the circumferential direction with a phase α, and the sensor signal obtained by the measurement is passed through the A / D converter. Two digital data synchronized by discretization are obtained, the digital data is subjected to Fourier transform, and vibration values F (m) and F _α (m) of order m of angular velocity Zω _c obtained by Fourier transform are used. Based on the following formula:
A _mZ-1 e ^{−j θ} = {e ^{j α} F (m) −F _α (m)} / 2 ^{j sin} _α
A _{mZ + 1} e ^{j θ} = {F _α (m) −e ^{−j α} F (m)} / 2 ^{j sin} _α
(Where m is the order of vibration, Z is the number of rolling elements, ω _c is the angular velocity of revolution of the rolling elements, and θ is the center between an unknown reference position on the fixed wheel and one of the vibration measuring sensors. Angle, F (m) is vibration value of order m of one of the vibration measurement sensors , and F _α (m) is the other vibration measurement sensor arranged at a phase α with respect to the one vibration measurement sensor. Vibration value of order m )
The unknowns A _mZ-1 e ^{−j θ} and A _{mZ + 1} e ^{j θ} are obtained, and the RMS values of the vibration components caused by the mZ-1 mountain and the mZ + 1 mountain are obtained from the unknowns by the following equations, respectively.
RMS value of mZ-1 peak component = √2 | A _mZ-1 e ^{−j θ} |
RMS value of mZ + 1 peak component = √2 | A _{mZ + 1} e ^{j θ} |
A method for evaluating the rotational accuracy of a rolling bearing, wherein the rotational accuracy is evaluated from these RMS values. The maximum amplitude value and the minimum amplitude value of the respective maximum RMS value of the angular velocity _{mZω c = √2 (| A mZ} -1 | + | A mZ + 1 |)
Minimum RMS value = √2 || A _mZ-1 | − | A _{mZ + 1} ||
The rotational accuracy evaluation method for a rolling bearing according to claim 1, wherein The phase of the maximum amplitude value and the minimum amplitude value of the angular velocity mZω _c is | F (m) | ² cos 2α + | F _α (m) | ² − {F (m) F ^* _α (m) + F ^* (m) F _{When α} (m)} cos α ≦ 0, if | γ ₀ | ≦ π / 4, the maximum is γ ₀ and γ ₀ + π and the minimum is γ ₀ ± π / 2, and π / 4 <| γ ₀ | ≦ π / 2 is minimum at γ ₀ and γ ₀ + π and maximum at γ ₀ ± π / 2,
| F (m) | ² cos 2α + | F _α (m) | ² − {F (m) F ^* _α (m) + F ^* (m) F _α (m)} cos α> 0, | γ ₀ | ≦ If π / 4, the minimum is γ ₀ and γ ₀ + π and the maximum is γ ₀ ± π / 2. If π / 4 <| γ ₀ | ≦ π / 2, the maximum is γ ₀ and γ ₀ + π, and γ ₀ ± smallest at [pi / 2 (provided that, 2 [gamma ₀ is given et is the following equation ^{(28), F * (m} ) is the complex conjugate of ^{_{F (m), F * α}} (m) is F _The method according to claim 2, wherein _α is a complex conjugate of (m) .

Using the vibration value of the fixed ring,

The RMS value of the vibration component caused by the mZ-1 peak and the mZ + 1 peak when the rotating wheel and the fixed wheel are used reversely according to the above equation (31) is the RMS value of the mZ-1 peak component = √2 | B _{mZ -1} |
RMS value of mZ + 1 peak component = √2 | B _{mZ + 1} |
The maximum amplitude value and the minimum amplitude value of the angular velocity mZω _c are set to the maximum RMS value = √2 (| B _mZ-1 | + | B _{mZ + 1} |)
Minimum RMS value = √2 || _BmZ-1 |-| _{BmZ + 1} ||
Where the absolute values of mZ-1 and mZ + 1 order Fourier expansion coefficients B _mZ-1 e ^{-j θ} and B _{mZ + 1} e ^{j θ} obtained from the Fourier transform are expressed as | B _mZ-1 | and | B _{mZ + 1} |) . The method for evaluating the rotational accuracy of the rolling bearing according to claim 1. When another vibration sensor is provided in the phase β, the following equations (32) to (34) are established,
A _mZ-1 e ^{−j θ} + A _{mZ + 1} e ^{j θ} + D _mZ e ^{−jmZ θ} = F ′ (m) (32)
A _mZ-1 e ^{−j ( θ + α )} + A _{mZ + 1} e ^{j ( θ + α )} + D _mZ e ^{−jmZ ( θ + α )} = F ′ _α (m) (33)
A _mZ-1 e ^{−j ( θ + β )} + A _{mZ + 1} e ^{j ( θ + β )} + D _mZ e ^{−jmZ ( θ + β )} = F ′ _β (m) (34)
^{Here, Δ = e j (α -mZ} β) + e -j (β + mZ α) -e -j (α + mZ β) -e j (β -mZ α) + e -j (α - β) - Assuming that ej ^{( α − β )} ≠ 0, the amplitude of each component can be calculated from the following equations (35) and (36).
| A _mZ-1 | = | A _mZ-1 e ^{−j θ} | = | {(e ^{j ( α -mZ β )} −e ^{j ( β −mZ α )} ) F ′ (m) + (e ^{j β} − e ^{−jmZ β} ) F ′ _α (m) + (e ^{−jmZ α} −e ^{j α} ) F ′ _β (m)} / Δ | (35)
| A _{mZ + 1} | = | A _{mZ + 1} e ^{−j θ} | = | {(e ^{−j ( β + mZ α )} −e ^{−j ( α + mZ β )} ) F ′ (m) + (e ^{− jmZ β} −e ^{−j β} ) F ′ _α (m) + (e ^{−j α} −e ^{−jmZ α} ) F ′ _β (m)} / Δ | (36)
RMS value of vibration component caused by mZ-1 peak and mZ + 1 peak respectively, RMS value of mZ-1 peak component = √2 | A _mZ-1 |
RMS value of mZ + 1 peak component = √2 | A _{mZ + 1} |
The maximum amplitude value and the minimum amplitude value of the respective maximum RMS value of the angular velocity _{mZω c = √2 (| A mZ} -1 | + | A mZ + 1 |)
Minimum RMS value = √2 || A _mZ-1 | − | A _{mZ + 1} ||
The method for evaluating the rotational accuracy of a rolling bearing according to claim 1. The maximum amplitude value and the minimum amplitude value of the phase of the angular velocity MZomega _c is the respective following formula (37)
D _mz e ^{−jmZ θ} = {(e ^{−j ( α − β )} −e ^{j ( α − β )} ) F ′ (m) + (e ^{−j β−} e ^{j β} ) F ′ _α (m) + ( e ^{j α} −e ^{−j α} ) F ′ _β (m)} / Δ (37)
When F (m) = F ′ (m) −D _mz e ^{−jmZ θ} and F _α (m) = F ′ _α (m) −D _mz e ^{−jmZ ( θ + α )} , | F (m) | ² cos2α + | F _α (m) | ² − {F (m) F ^* _α (m) + F ^* (m) F _α (m)} cos α ≦ 0, if | γ ₀ | ≦ π / 4 up to ≦ [pi / 2 if minimum gamma ₀ and gamma ₀ + [pi and _{γ 0 ± π / 2 | γ} 0 and a maximum and at a minimum at γ ₀ ± π / 2 at _{γ 0 + π, π / 4} <| γ 0 And
| F (m) | ² cos 2α + | F _α (m) | ² − {F (m) F ^* _α (m) + F ^* (m) F _α (m)} cos α> 0, then | γ ₀ | ≦ If π / 4, the minimum is γ ₀ and γ ₀ + π and the maximum is γ ₀ ± π / 2. If π / 4 <| γ ₀ | ≦ π / 2, the maximum is γ ₀ and γ ₀ + π, and γ ₀ ± (2γ ₀ is given by the following equation (28) , F ^* (m) is a complex conjugate of F (m), and F ^* _α (m) is F _α 6. The method for evaluating the rotational accuracy of a rolling bearing according to claim 5, wherein (m) is a complex conjugate .

  A rotational accuracy measuring device for a rolling bearing, wherein the rotational accuracy of the rolling bearing is measured using the rotational accuracy evaluation method according to claim 1.

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