Influence of geometric defects in bearing outer race on vibration generation: an oriented study for manufacturing tolerances specification and allocation.

Serrano Mira, Julio ; Bruscas Bellido, Gracia M. ; Abellan Nebot, Jose Vcte 等

1. INTRODUCTION

To find out how geometric defects affect mechanism functionality it is very important to establish the types of defects to be controlled and their permissible values. The analysis must ensure part function, permitting maximum variability in manufacturing and thereby reducing costs.

This paper analyses the influence of waviness in bearing raceway outer ring on vibration generation, finding a relationship that allows direct control of the functional requirement "vibration" during manufacturing stage, rather than making indirect control through geometric tolerances with no distinction between magnitude and type of waviness. This analysis suggests that only waves of certain orders generate significant vibrations, thus being this one the geometric parameter to control, though usually done in an indirect way by specifications of roundness.

2. PROBLEM STATEMENT

A functional requirement to be considered during bearing design is the limitation of generated vibrations, in order to reduce noise and to avoid resonance effects by natural frequencies coupling of other mechanism components. Hence, control of clearances and geometric defects of contact surfaces is important. Usually, tight dimensional and geometric tolerances are used to ensure a better functionality. However, such restrictive tolerances often entail an indirect control that hides the real problem origin, though solving it. In order to reduce error magnitude, better machines and more controlled manufacturing processes will be required.

Vibrations have three main characteristics: frequency, wave length and wave amplitude. The objective of this work is to determine the necessary geometric functional constraints (GFC) in the outer race of a bearing with angular contact to fulfil the functional requirement of vibrations absence. With this purpose, dependence of generated vibrations magnitude and its frequency with eventual irregularities present in the bearing outer race are analysed. The result will allow establishing functional relationships or equations (FEq) between the functional requirement (FR) "vibration" (frequency and amplitude) and the geometric feature "profile defects" of the analysed element (Figure 1). The case of defects in the inner race is analysed in previous work (Serrano & Sancho, 2000).

[FIGURE 1 OMITTED]

3. ANALYSIS OF RESULTING VIBRATIONS

In bearing components, waviness due to imperfect manufacture is an imperfection whose wavelength is much bigger that contact area width between balls and raceways (Hertzian contact width). Thus, it can be assumed that contact local deformation does not influence on waviness profile.

Waviness effect on resulting vibrations has been studied by many authors (Yhland, 1967; Harris & Kotzalas, 2007). In the work by Akturk (Akturk, 1999), vibrations produced by waviness in the inner and outer raceways and in the balls are studied, modelling the shaft-bearing set as a mass-spring system, where the shaft acts as a mass and the raceway and balls as massless non-linear springs. In this way, the system undergoes non-linear vibrations under dynamic conditions.

The relationship between the i-th ball ([[delta].sub.i]) and the Hertz contact force ([W.sub.i]) is obtained according to [W.sub.i]=Kx[[delta].sup.3/2.sub.i], where K is the stiffness coefficient for the same material properties of the two contacting bodies K=[([K.sup.-2/3.sub.i] + [K.sup.2/3.sub.o]).sup.-2/3], where [K.sub.i] and [K.sub.o] are inner and outer raceways to ball contact stiffness respectively (Harris & Kotzalas, 2007; Akturk, 1993).

[FIGURE 2 OMITTED]

Figure 2 illustrates the analysed ball bearing and a view of the defects in the outer raceway. It can also be seen that a sinusoidal waviness of amplitude [C.sub.p] is considered and also a constant interference due to a preload of amplitude Co. Assuming that the inner race moves at the speed of the shaft [[omega].sub.i] and the ball centre at the speed of the cage [[omega].sub.c], the height of the waviness to consider as the interference between the i-th ball and the outer race can be expressed as a function of time [C.sub.i]=[C.sub.o]+[C.sub.p] x sin[N x(9+([[omega].sub.c]-[[omega].sub.i]) x t + [gamma]i)], where 3 is the angle between the ball number 0 and the reference axis, N is the number of waviness on the circumference, and [gamma] is the angle between consecutive balls.

The analysis has been carried out for the system described in Table 1, assuming in the model that the shaft is perfectly rigid and uniform, and it is supported by two preloaded angular ball bearing (15[degrees] contact angle).

Applying the movement equations to this system and solving them for this particular case, using the iterative Runge-Kutta method, results are showed in Figure 3, and it can be concluded that:

1) For vibrations with greater amplitude, the frequency depends on the waviness order according to the relations: k=q*m[+ or -]p (waviness order) and qmmc (frequency for vibrations caused), where m is the number of bearing balls, and p and q are integers [greater than or equal to]1 and [greater than or equal to]0 respectively. However, vibrations of smaller amplitude appear also at other frequencies.

2) The most severe vibrations appear when Ball Passage Frequency (BPF) matches natural frequency of the system.

3) The most severe vibrations appear for a waviness order k=im[+ or -]1, in the radial vibration case, and k=im in the axial vibration case. For some order waviness, the amplitudes of vibrations are negligible.

Previous results evidence that the greatest vibration amplitudes appear for those order waviness included between a preceding and following multiple of the balls number.

[FIGURE 3 OMITTED]

4. OBTAINING GEOMETRICAL CONSTRAINTS

From previously analysed results, functional geometrical constraints (FGC) and their relationships to functional requirements (FR), that is, functional equations (FEq), can be established. First, maximum possible information influencing geometrical characteristics is extracted and then how these, in turn, influence FR. It can be seen that:

1) Resulting vibration frequencies depend on the rotating speed and on the relationship between the balls number and the waviness order. Thus, a concordance relationship can be established.

2) Vibrations severity depends on the waviness order and, to a smaller extent, on the waviness amplitude.

3) It is important to avoid those waviness orders which are a preceding and following multiple of a balls number. For other combinations, vibrations are negligible. E.g., in a 6-8 balls bearing, common low order defects that lead to elliptical, triangular or square forms, produce very small amplitude vibrations.

It can be noticed that vibrations frequency and amplitude depend on the waviness magnitude and order. Therefore, the FGCs are: FGC1 (roundness error) and FGC2 (waviness order), existing three functional equations which relate their value with deformation and vibrations.

* FGC1: Roundness tolerance of the outer raceway. Its value is limited to allow a uniform running and to avoid balls blocking or excessive races and balls deformations, and the incomplete contact between race balls and races. The condition is FGC1<#, where # is a specific value.

* FGC2: Waviness types to avoid in the outer raceway. Waviness (periodical) that produces great amplitude vibrations should be avoided.

* FEq1: Relates the deformation of the races and of the balls contact area when a force is applied. It is the hertzian contact mentioned in section 3.

* FEq2: Relates the resulting vibration amplitude and the waviness order. The most severe vibrations appear for certain waviness orders. Therefore, it limits configurations which generate great amplitudes. The FGC2 leads to k [not member of] [i x m - 1, i x m + 1 ], where i=1, 2, 3, ...

* FEq3: Relates resulting vibration frequencies with the rotating speed and waviness orders. It establishes what frequencies are produced and they can be used, if necessary, to limit or avoid a specific frequency spectrum. The FGC2 leads to f = qxmx[w.sub.c]/2x[pi].

5. CONCLUSIONS

From the analysis carried out, dependence between generated vibrations and type and magnitude of raceway imperfections has been obtained. Accordingly, geometric parameters to be controlled to limit undesirable vibrations have been deduced, namely: roundness error and waviness order. Typically, just a roundness tolerance is considered, thus being very restrictive to avoid, indirectly, all originated vibrations. However, these tolerances could be wider if waviness order would also be controlled to avoid really harmful frequencies.

These geometric parameters make it possible to take actions directly in the origin of incorrect function. Transference of the maximum functional information to the manufacturing stage allows process optimization with assurance of required function. In this way, the two maxims of Functional Geometric Dimensioning and Tolerancing (FGD&T) would be satisfied: every part meeting FGD&T can be used, and parts that can be used will not be rejected for not meeting FGD&T.

6. REFERENCES

Akturk, N. (1993). Dynamics of a rigid shaft supported by angular contact ball bearings, PhD Thesis, Imperial College of Science, Tech. & Medicine, London

Akturk, N. (1999). The effect of waviness on vibrations associated with ball bearing, Journal of Tribology, Vol. 121, October 1999, pp. 667-677, ISNN: 0742-4787

Harris, T.A. & Kotzalas, M.N. (2007). Rolling Bearing Analisys, CRC Press, ISBN: 084937183X, Boca Raton

Serrano, J. & Sancho, J. (2000). Analisis de la influencia de defectos geometricos en los anillos de un rodamiento en la generation de vibraciones, orientado a la asignacion de tolerancias de fabrication, Proc. of the XIV Cong. Nac. Ing. Mecdnica, Diaz, V.; Garcia, J.C. & San Roman, J.L. (Ed.), pp. 461-466, ISBN: 0212-5072, Leganes, December 2000, Universidad Carlos III, Madrid

Yhland, E. M. (1967). Waviness measurement. An instrument for quality control in rolling bearing industry, Proc. IMechE, Vol. 182, Part 3K, pp. 438-445

Tab. 1. Data of the analysed system

Inner ring bore diameter: 40 mm
Inner ring diameter: 46 mm
Outer ring diameter: 62 mm
Inner ring groove radius: 4,1mm
Outer ring groove radius: 4,6mm
Ball diameter ([d.sub.b]): 8 mm

Number of balls: 8 u.
Unloaded contact angle: 15[degrees]
Pitch diameter of ball set: 54 mm
Mass of the shaft: 550 N
Preload each ball: 10 N
Shaft rotating speed (rpm): 5000