2.1 Load-Deflection Relationships of a Rolling Bearing
Generally, when a tapered roller bearing with contact angle $\alpha$ is subjected simultaneously to a radial and an axial load, Fig. 1 shows the relative radial displacement $\delta_r$ and the axial displacement $\delta_a$ between bearing rings [1]. At any angular position $\psi$ measured from the most heavily loaded rolling element, the approach of the rings can be expressed as
$\delta_\psi=\delta_a \sin \alpha+\delta_r \cos \alpha \cos \psi$
(1)
The maximum relative deflection at $\psi=0$ is given by
$\delta_{\max }=\delta_a \sin \alpha+\delta_r \cos \alpha=\frac{2 \varepsilon}{2 \varepsilon-1} \delta_a \sin \alpha$
(2)
where $\varepsilon$ is the load distribution factor
$\varepsilon=\frac{1}{2}\left(1+\frac{\delta_a \tan \alpha}{\delta_r}\right)$
(3)
In condition of pure axial deflection for standard tapered roller bearings made of the bearing steel [1], the axial deflection can be approximated as
$\delta_{a 0}=\frac{0.000077}{\sin \alpha} \frac{Q_{\max }^{0.9}}{l_e^{0.8}}=\frac{\delta_{\max }}{\sin \alpha}$
(4)
where $Q_{\max }$ is the maximum roller normal load $(\mathrm{N})$, and $l_e$ is the effective roller length (mm).
Figure 1. Rolling bearing deflection due to combined radial and axial loading
From eqns (2) and (4), the axial deflection $\delta_a$ can be derived as
$\delta_a=\frac{0.000077}{\sin \alpha} \frac{Q_{\max }^{0.9}}{l_e^{0.8}} \frac{2 \varepsilon-1}{2 \varepsilon}.$
(5)
Using load integrals, applied axial bearing load can be expressed as
$F_a=Z \sin \alpha Q_{\max } J_a(\varepsilon)$
(6)
where Z is the number of rollers. J is defined as
$J_a(\varepsilon)=\frac{1}{2 \pi} \int_{-\psi_l}^{\psi_l}\left[1-\frac{1}{2 \varepsilon}(1-\cos \psi)\right]^{1.11} d \psi.$
(7)
Applied radial bearing load can be expressed as
$F_r=Z \cos \alpha Q_{\max } J_r(\varepsilon)$
(8)
$J_r(\varepsilon)=\frac{1}{2 \pi} \int_{-\psi_l}^{\psi_l}\left[1-\frac{1}{2 \varepsilon}(1-\cos \psi)\right]^{1.11} \cos \psi d \psi$
(9)
$\psi_l=\cos ^{-1}(1-2 \varepsilon).$
(10)
Consequently, if bearing loads $F_a$ and $F_r$ are given, $Q_{\max }$ and $\varepsilon$ can be calculated numerically from eqns (6) and (8). Using them, $\delta_r$ and $\delta_a$ can be acquired from eqns (3) and (5).
2.2 Equilibrium Conditions of a Tapered Roller Bearing Unit
Figure 2 shows a typical tapered roller bearing unit used in railway vehicles. From the SKF guide principles, which are widely used in railway industry [6, 7], the equivalent radial axlebox load can be calculated as
$K_r=f_0 f_{r d} f_{t r} G$
(11)
where $f_0$ is the payload factor, $f_{r d}$ is the dynamic radial factor, $f_{t r}$ is the dynamic traction factor, and $G$ is the axlebox load.
The equivalent axial axlebox load can be calculated as
where $f_{a d}$ is the dynamic axial factor.
Considering the loading conditions, the following forces and moment equilibrium equations can be derived.
$K_r-F_{r o}-F_{r i}=0$
(13)
$K_a+F_{a o}-F_{a i}=0$
(14)
$l_d K_a+l_c F_{r o}-\frac{l_c}{2} K_r=0$
(15)
where $l_d$ is the axial axlebox load distance (mm) (Fig. 2), and $l_c$ is the distance between bearing load centres (mm).
Figure 2. Axle bearing unit with applied loads
In order to obtain the four unknown bearing loads, one more equation can be used from the geometric constraint which is assumed that total axial clearance is unchanged after loading.
$\delta_{a o}+\delta_{a i}=\delta_0$
(16)
where $\delta_0, \delta_{a o}$, and $\delta_{a i}$ are the initial axial clearance, the outer bearing axial deflection, and the inner bearing axial deflection, respectively.
All bearing loads can be obtained by solving eqns (13)–(16).
2.3 Fatigue Life of a Tapered Bearing Unit
The fatigue life of a rolling bearing is calculated from the following formula:
$L_{10}=\left(\frac{C}{P}\right)^{10 / 3} \times 10^6(\mathrm{rev.)}.$
(17)
where C is the basic dynamic capacity of a rolling bearing (N).
The equivalent dynamic bearing load P is
$P=\frac{J_r(0.5) J_1(\varepsilon)}{J_1(0.5) J_r(\varepsilon)} F_r$
(18)
$J_1(\varepsilon)=\left\{\frac{1}{2 \pi} \int_{-\psi_l}^{\psi_l}\left[1-\frac{1}{2 \varepsilon}(1-\cos \psi)\right]^{4.4} d \psi\right\}^{1 / 4}.$
(19)
Because a tapered roller bearing unit has two bearings, the total fatigue life of a bearing unit is given by
$L_{10_{\_} \text {tot }}=\left(L_{10_{\_} \text {in }}^{-1.125}+L_{10_{\_} \text {out }}^{-1.125}\right)^{-8 / 9}.$
(20)
Open Access
,
View Full Article|
Download PDF
Views: 202
Crossref: 0