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Vibration Analysis-Based Fault Diagnosis of a Dynamically Loaded Bearing with Distributed Defect

  • RESEARCH ARTICLE - SPECIAL ISSUE - Impact of Advanced Tribological Technologies on Modern Industry
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Abstract

This research paper presents the analysis of vibration generated due to the race waviness of rolling element bearing which is under external dynamic load. The waviness on races is modeled either by sinusoidal function or superposition of sinusoidal functions of different orders. The mathematical model developed to analyze elastic deformations at race-rollers contacts and nonlinear contact forces for a defect free bearing under static external loading is modified and extended to take dynamic load into account. The radial waviness on races at these contacts is introduced in order to determine un-balanced excitations causing the bearing to vibrate. The set of equations that represent the dynamics of system have been obtained by inputting the excitations to linear vibratory model of bearing system. The vibration response of bearing has been obtained by numerical simulation of state space model representing dynamics of bearing system. The spectra of vibration response are analyzed for different amplitude contribution of dynamic component of external load. The numerical results have been obtained for NJ204 bearing with known values of order and amplitude of waviness. For harmonic loading, the additional spectral components have been observed for each order of waviness on both outer and inner race. The additional spectral components have also been observed for outer race waviness modeled as superposition of different order waviness models, whereas for inner race, there is some amplitude variation in spectral components. The vibration frequencies resulting from various orders of waviness on races have been validated with those of prior researchers.

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Abbreviations

A :

Amplitude of harmonic loading due to unbalanced mass

[A]:

State matrix

b :

Damping coefficient of element

[B]:

Input matrix

C d :

Load deformation constant

[C]:

Output matrix

d :

Diameter of roller

D :

Bearing pitch diameter

[D]:

Direct transmission matrix

E :

Modulus of elasticity

E′ :

Effective modulus of elasticity

f id :

Inner race ball passing frequency

f od :

Outer race ball passing frequency

f s :

Shaft frequency

F :

External radial load

h f :

Minimum film thickness

I :

Moment of Inertia of cross section of race

j :

jth revolution

J r :

Radial integral

K :

Stiffness of element

k :

kth rolling element from upward line of action of load

L :

Length of roller

\(m\) :

Order of waviness

M :

Mass of element

\(n\) :

nth rolling element from downward line of action of load

P d :

Diametral clearance of bearing

P :

Contact force

P max :

Maximum contact force

Q :

Input excitations

R :

Radius of neutral axis

R x, race :

Effective radius of roller and race contact in the direction of motion

t :

Instantaneous time

u :

Mean surface velocity relative to lubricated conjunctions in the direction of motion

\(w\) :

Magnitude of waviness

W :

External static loading

\({\text{WO}}\) :

Arbitrary value of waviness order

x :

State variable

X :

Dimensionless stiffness coefficient of fluid film

y :

Displacement of mass

Y :

Dimensionless damping coefficient of fluid film

z :

zth mode of flexural vibration

Z :

Number of rolling elements

\(\beta\) :

Amplitude of waviness

\(\varphi\) :

Phase difference

δ :

Contact deformation

\(\vartheta\) :

Load-defection coefficient

ψ l :

Extent of load zone

є :

Load distribution factor

є o :

Eccentricity ratio

\(\omega\) :

Angular frequency

ω nat :

Natural frequency

η :

Lubricant viscosity at bearing operating temperature at atmospheric pressure

ξ :

Lubricant viscosity pressure coefficient

λ k :

A constant

c :

Cage

ci:

Contact waviness of roller with inner race

co:

Contact waviness of roller with outer race

f :

Lubricating fluid

i :

Inner race

if:

Lubricant fluid for the conjunction of inner race and roller

l :

Line of action of external load

o :

Outer race

of:

Lubricant fluid for the conjunction of outer race and roller

r :

Rolling element

s:

Shaft

T :

Transpose

˙:

First derivative

¨:

Second derivative

{}:

A vector

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Acknowledgements

This research work is presented at International tribology research symposium 2020 (ITRS 2020) on impact of tribology on society, conducted during 5th to 7th November 2020. The article ID is ITRS106.

Funding

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. Faculty of Science and Technology, ICFAI Foundation for Higher Education, Hyderabad, 501203, India

    Govardhan Tingarikar

  2. Bhartiya Skill Development University, Jaipur, 302037, India

    Achintya Choudhury

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  1. Govardhan Tingarikar
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Correspondence to Govardhan Tingarikar.

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Tingarikar, G., Choudhury, A. Vibration Analysis-Based Fault Diagnosis of a Dynamically Loaded Bearing with Distributed Defect. Arab J Sci Eng 47, 8045–8058 (2022). https://doi.org/10.1007/s13369-021-05862-7

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