The ball bearing comprises the outer ring, inner ring, cage and rolling elements. A standard bearing achieves dynamic balance in a stable operating condition, while a series of impulses will be generated once there is a defect between the contact surfaces. A 5-DoF dynamics model and a defect model will be introduced in the following.
2.1. The Nonlinear 5-DoF Model
This model describes the nonlinear dynamic behavior of bearing, as shown in
Figure 1. In the 5-DoF model, 4 DoF represents the horizontal and vertical directions of inner and outer rings, and 1 DoF stands for the vertical direction of the resonator, which is modeled as the spring-mass system [
10].
Based on Newton’s second law, the bearing dynamic equilibrium equations can be formulated as Equation (
1) [
10].
and
are contact force at
x and
y axis, respectively,
is external load. The meanings and values of other variables can be found in [
15]. According to Hertzian contact theory, the contact force between rolling element and raceways can be given by:
with
j from 1 to
,
which represents the number of rolling elements.
stands for the ball’s stiffness,
denotes deformation. The deformation of the
jth ball,
, is determined by the displacement between the inner and outer races, the angular position
and the total clearance
c caused by the oil film and assembly clearance, as:
The angular position of the
jth ball can be calculated by Equation (
4).
where
is initial cage angular position and
is cage angular frequency, which can be further obtained from shaft frequency
like Equation (
5).
where
and
are the ball diameter and pitch diameter, respectively.
Usually, there is inevitable sliding for bearings in real applications when a ball rolls on the raceways. The sliding direction depends on where the ball is located; when the ball enters the load zone, the angular speed of the ball center is faster than that of the cage; otherwise, the ball would slide backward. Consequently, with sliding considered, the angular position of each ball can be modified by Equation (
6) [
10].
There are two constants and a sign function in Equation (
6).
is a parameter defining the mutation percentage of average contact frequency, which is normally between 0.01 and 0.02 rad.
is a random number with uniform distribution with the range of
, and the sign function
is expressed as:
Considering
should be nonnegative in physics, its final value is determined by:
With the contact force of each ball obtained from Equation (
2), the total contact forces in
x and
y direction are determined with Equations (
9) and (
10).
2.2. Bearing Defect Model
When bearing has defects either on races or balls, an additional deformation,
, will be released when the ball moves over the defect zone. Thus, with defect considered, deformation of the
jth ball can be further identified as:
Once bearing deformation under defect is obtained, it can be substituted into Equations (
9) and (
10). The nonlinear contact force can be calculated and further substituted into Equation (
1) for the fault-bearing response.
changes with defect position, defect shape and the number of defects. Due to space limitation, only the modeling of defect position will be presented in this paper, the modeling of defect size, defect shape, and multiple defects can be found in the original conference paper [
15].
Before defect modeling, four basic geometrical parameters are chosen to feature the defect, as demonstrated in
Figure 2, the defect width
B, the defect depth
, the defect initial angle
and the defect span angle
. Take a defect on the outer ring as an example, the relation between
and
B is expressed as:
Suppose the local defect depth is
, the additional deformation
generates only when a ball falls into the defect zone within
and
, then the deformation released by defect on raceway is given by:
The defect location on the outer ring or inner ring changes with different rules. For the outer ring, the defect is fixed at the defect initial angle
. However, for the inner ring, the defect location changes with time when the inner ring rotates. Thus,
in Equation (
13) can be further modeled as follows.
Different from rings, when a defect happens on a rolling element, the defect spins with ball speed
and its position
can be obtained like:
hereby the ball speed
can be calculated from shaft speed as follows,
The defect on balls contacts the inner and outer ring periodically. Besides, the curvature radii of inner and outer rings are different. Therefore, the same defect span angle
produces different angular widths. The angular widths of defect on the outer ring and inner ring,
and
, can be calculated by Equations (
17) and (
18).
with
and
as the diameters of outer ring and inner ring, respectively.
Obviously, the curvature radius determines the depth when ball enters into the raceway, and the curvature radii of ball
, inner ring
and outer ring
can be obtained, respectively, by Equations (
19)–(
21) [
16].
During one revolution of the ball, defect contacts the inner ring and outer ring in succession, with an angular distance of
. So,
can be given by Equation (
22) [
16].
Deformation released by fault only appears on the fault ball (
kth ball). As a result, the contact deformation with ball defect is given by:
Besides the test bearing, the virtual bearing test bench also consists of a driving module and a loading module. The loading module guarantees that test bearing works under defined external load, and the driving module is responsible for speed profile definition. In this research, the driving module is modeled by a DC motor and a shaft, while the loading module is modeled by an electro-hydraulic servo system. More details about the modeling of these two modules are introduced in [
15].