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An experimental testing of a simplified model of a ball bearing: stiffness calculation and defect simulation

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Abstract

This paper presents an experimental verification of a simplified model of a nonlinear stiffness ball bearing in both static and dynamic modes and testing its capabilities to simulate accurately fault’ effects.

Analytical model was developed using a different method comparatively to classical ones; the ball’s deformation is obtained without using Palmgren’s method. Modeling considers the balls scrolling in the cage and the effect of the load-rotating vector. Obtained formula allows introduction of defects characteristics under parametric forms.

These modifications were done in order to realize two objectives. The first one is to ameliorate ball bearings stiffness computing and to improve a more realistic simulation of dynamic behavior of a defective ball bearing. The second is to use obtained results in design and maintenance domain.

To verify experimentally the developed model of rigidity in both static and dynamic conditions, a number of compression tests were done on the ball bearings. This requires the specimen grips system adaptation of a mechanical universal testing machine to receive the non-usual specimen. A number of experimental simulations of the main faults are done on a testing bench to verify the defect model.

Results of defects simulation and model behavior in statics and dynamics are compared to experimental results. The developed model gives an acceptable similarity and it proves its simplicity and robustness. Both results were acceptable.

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Abbreviations

α::

angle, fault position,

β::

contact angle of ball with ring,

δ i ::

deflection/deformation,

δ T ::

total deformation,

γ::

train (cage) angular rotation,

ε::

radius of curve of ball’s housing,

ϕ::

angle /x, ball’s position /x

ϕ n ::

gap angle between two balls,

ν::

Poisson’s ratio,

θ::

inner ring tilt angle /z,

ω::

shaft angular or spin speed,

ω ca ::

cage rotational speed,

Δ::

fault size,

φ::

load angular position,

ψ::

angle, inner ring tilt angle /y

a::

half contact width,

b::

width,

c::

damping coefficient,

d b ::

ball diameter

d ou ::

outer ring diameter in ball housing,

d in ::

inner ring diameter in ball housing,

d i ::

displacement /R, X, y and Z,

d log ::

ball housing diameter,

d p ::

bearing pitch diameter,

h::

gap between rings,

h r ::

local deformation depth,

L c ::

contact width of the (ball/ring),

n::

ball number,

m::

loaded balls number, mass,

p::

ball bearing perimeter,

q 0::

maximum distributed loading,

q(p)::

distributed loading along perimeter p,

q(r)::

distributed loading along r,

q(z)::

distributed loading along z,

r m ::

bearing pitch radius,

A r ::

elementary area according to the axis r,

E i ::

Young’s module for (ball, ring),

F i ::

force /r, t, x, y & z,

F m ::

maximum force,

F r ::

radial force,

K rr ::

radial stiffness,

M::

mass, moment,

NR::

shaft speed

NT::

cage speed

NB::

ball speed

R, R b ::

ball radius,

R a ::

shaft radius,

R 1e ::

external inner ring radius,

R 2i ::

internal outer ring radius,

R log ::

ball’s housing radius,

R b ::

radius ball,

T::

cycle time,

W::

load rotating vector.

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Authors and Affiliations

  1. COMENA/CRNB/DTN/Laboratoire des Etudes Mécaniques, CRNB, B.P. 180, Ain-Oussera, 17200, Willaya de Djelfa, Algeria

    M. Dougdag, N. E. Titouche & M. Djaoui

  2. Département de Mécanique, Faculté des Sciences Pour l’Ingénieur, Université Saad Dahlab Blida, B.P. 27, Route de Soumâa, Blida, Algeria

    M. Ouali

  3. COMENA/CRND, Draria, Algeria

    H. Boucherit

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  1. M. Dougdag
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  2. M. Ouali
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  4. N. E. Titouche
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  5. M. Djaoui
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Correspondence to M. Dougdag.

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Dougdag, M., Ouali, M., Boucherit, H. et al. An experimental testing of a simplified model of a ball bearing: stiffness calculation and defect simulation. Meccanica 47, 335–354 (2012). https://doi.org/10.1007/s11012-011-9434-0

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