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Vibration analysis of dynamically loaded bearing with distributed defect based on defect induced excitation

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Abstract

In the present study an investigation has been carried out to determine distributed defect induced excitations in a bearing subjected to an external load which is a combination of static and dynamic load. The dynamic component of external load has been considered to be harmonic in nature and may be due to the shaft unbalance. The race surface waviness in radial direction has been taken in to account as the distributed defects in bearings. The race waviness is modeled mathematically, as sinusoidal functions of different orders and superposition of sinusoidal functions as well. Equal and random amplitudes of different orders have been considered. Kinematic and dynamic parameters as well as contact forces resulted due to race-rolling element interaction as obtained from published literature have been modified and extended to consider radial waviness and external dynamic loading. The frequency spectra of nonlinear contact forces causing excitations have been obtained. For static radial loading, these spectra are in accordance with those obtained for responses of linear dynamic models. For external dynamic loading, however, additional spectral components as sidebands at shaft frequency about the significant frequencies corresponding to external static loading have been observed for both inner and outer race surface waviness, when modeled as sinusoidal function of any order. Additional spectral components have also been observed for outer race waviness when formulated as superposition of different orders whereas for inner race, there are some amplitude variations in spectral components compared to that of static loading. Variation in amplitudes of spectral components has also been resulted when random values were assigned to different orders of waviness.

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Abbreviations

a :

A whole number

A :

Amplitude

b :

A whole number

d :

Diameter of rolling element

D :

Pitch diameter of bearing

e :

Excitation at each race roller contact in radial direction

E :

Total excitation in vertical direction

\(f_{id}\) :

Ball passing frequency inner race

\(f_{od}\) :

Ball passing frequency outer race

\(f_{s}\) :

Shaft frequency

\(F_{r}\) :

Radial loading

\(J_{\mathrm{r}}\) :

Radial integral

K :

Load-deformation constant

L :

Effective length of roller

m :

Order of waviness

n :

\(\hbox {n}\mathrm{th}\) revolution

p :

An integer

P :

Contact force in a defect free bearing

\(p_{d}\) :

Diametral clearance of bearing

\(P_{w}\) :

Net contact force after considering contact waviness

q :

An integer

r :

Groove radius

R :

Referred rolling element when load acting downwards

S :

Rolling element immediate from direction of load when load acting upwards

u :

Magnitude of race waviness

t :

Instantaneous time

w :

Waviness in contact with roller

W :

Static load

X :

Number of rolling elements from reference roller, \(\hbox {R}_{1}\) to meet \(\hbox {S}_{1}\)

WO :

Any arbitrary value of waviness

Z :

No. of rolling elements

\(\alpha \) :

Nominal contact angle

\(\omega \) :

Circular frequency

\(\beta \) :

Phase difference

\(\delta \) :

Elastic deformation at race roller contact

\(\vartheta \) :

Load-deformation index

\(\upkappa \) :

A constant

\(\varepsilon \) :

Load distribution factor

\(\pm \psi _{l}\) :

Extent of load zone

c :

Cage

h :

Harmonic load

i :

Inner race

l :

Line of action of external load

max :

Maximum

o :

Outer race

s:

Shaft

w :

After considering waviness

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Acknowledgements

This research did not receive any specific Grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Authors and Affiliations

  1. Sir Padampat Singhania University, Udaipur, 313601, India

    T. Govardhan & Achintya Choudhury

  2. Madhav University, Abu road, Sirohi, 307026, India

    Deepak Paliwal

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  1. T. Govardhan
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Correspondence to T. Govardhan.

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Govardhan, T., Choudhury, A. & Paliwal, D. Vibration analysis of dynamically loaded bearing with distributed defect based on defect induced excitation. Int. J. Dynam. Control 6, 499–510 (2018). https://doi.org/10.1007/s40435-017-0324-8

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