Civil Engineering Reference
In-Depth Information
FIGURE 1.7
Experimental and synthesized profiles: Grind-II.Da.
Performance Comparison: Axial Vibrations in Ball Bearings
The goal of this section is to bolster the validity of the fractal-wavelet method with an application to a
design example, namely the tolerance design for rolling elements in ball bearings. The salient features of
the design problem are introduced below.
Ball bearings are critical machine elements used to transmit power and to support loads. Typical
applications are machine tool spindles, power tools, etc. The specific problem under study concerns
bearings used to support axial loads, and the influence of form errors in the bearing elements on the
axial vibration force. These form errors induce Hertzian contact forces, which augment the axial force.
Large axial vibrations can lead to catastrophic failures.
Analysis of Bearing Forces
Figure 1.8
shows the components in an angular contact ball bearing. It is assumed that the applied load
to the bearing is purely a thrust load. The performance parameter of interest is the axial vibration force.
Additional assumptions made in calculating bearing forces are: the bearing has
N
b
equispaced balls, firmly
in contact with the inner and outer races; the rotational speed of the bearing is moderate, implying
negligible centrifugal forces; and the outer race is stationary. Geometric errors in the rolling elements
result in both axial and radial vibration forces [71].
In this section, the error is synthesized using the theory developed above and is superposed on the
circumference of the outer race. The inner race and the balls are considered to have ideal geometries.
The equations for the axial vibration force can be derived from the Hertzian contact forces. The normal
load
Q
j
on a ball
j
in terms of the corresponding deflection
j
is:
C
f
j
3/2
,
Q
j
1
j
N
b
,
(1.15)
j
0
where
C
f
is the contact compliance factor. The contact deflection
is influenced by the geometric errors
in the outer race, and is hence expressed as a nominal deflection
(i.e., in the absence of any geometric
error) and a time-varying component
j
(
t
), which is dependent on the relative position of the ball
j
, and
the outer race:
j
0
j
t
()
(1.16)
Equation (1.15) is approximated assuming a constant
C
f
[71] as below:
j
2
3
j
t
()
3
()
t
3/2
Q
m
1
--------------
---------------
(1.17)
2
0
2
8
