Model description
This model represents the dynamics of a roller bearing.
The frictional losses at the rotational speed $\omega$ are calculated as
\begin{equation}\label{eq:2_motorBearingQDot}
\dot{Q}_b=\omega\cdot M_b,
\end{equation}
where as $M_b$ is the frictional torque. To characterize the frictional torque, the approach by Harris and Kotzalas is used:
$$M_b=M_{b,1}+M_{b,2}$$
$M_{b,1}$ describes the frictional torque due to the applied load, while $M_{b,2}$ includes the torque due to viscose friction in the lubricant.
According to Palmgren, $M_{b,1}$ depends on the static equivalent load $F_s$, the static basic load rating $C_s$, the applied load $F_\beta$, the bearing diameter $d_b$ and the two empirical parameters $z$ and $y$:
\begin{equation}
M_{b,1}=z\cdot \left(\frac{F_s}{C_s}\right)^y\cdot F_\beta\cdot d_b
\end{equation}
The applied load depends on the axial and radial factor of the bearing - $X$ and $Y$, the contact angle $\alpha_b$ and the axial and radial force - $F_a$ and $F_r$ - acting on the bearing:
\begin{equation}
F_\beta = \max\left\lbrace X\cdot F_a\cdot \cot\alpha_b-Y\cdot F_{ro},\cdot F_{r}\right\rbrace
\end{equation}
For radial ball bearings, $X=0.9$ and $Y=0.1$ applies.
The factors for other bearing configurations are listed in DIN 281.
Harris and Kotzalas choose the empirical parameters $y$ and $z$ depend on $\alpha_b$ and the bearing type as shown in the following table:
Bearing Type | $\alpha_b$ | $z$ | $y$ |
Radial deep-groove | 0 | 0.0005 | 0.55 |
Angular-contact | 30-40 | 0.001 | 0.33 |
Thrust | 90 | 0.0008 | 0.33 |
Double-row, self-aligning | 10 | 0.0003 | 0.40 |
The contribution of viscose friction to the bearing torque can be estimated using the empirical formula by Palmgren
\begin{equation}
M_{b,2}=\begin{cases} 4.5\cdot 10^{-2}\cdot f_0\cdot \left(\nu_{lub}\cdot \omega\right)^{2/3}\cdot d_m^3 & \text{if } \nu_0\cdot n\geq210\cdot 10^{-6}\\
160\cdot 10^{-7}\cdot f_{0}\cdot d_m^3 & \text{else}
\end{cases}
\end{equation}
The values suggested for $f_0$ are shown in the following table:
Bearing Type | Grease | Oil Mist | Oil Bath | Oil Jet |
Cylindrical roller with cage | 0.8 | 2.1 | 3.1 | 3.1 |
Cylindrical roller full complement | 7.5 | - | 7.5 | - |
Thrust cylindrical roller | 9 | - | 3.5 | 8 |
The heat flux through all $N$ rollers from side 1 to side 2 is given as
\begin{equation}
\dot{Q}_{1\rightarrow2}=-\dot{Q}_{2\rightarrow1}=N\cdot \lambda_i(d_m, \omega)\cdot (T_1-T_2).
\end{equation}
$\lambda_i$ is the thermal resistance of a single roller and approximated using the approach by Weidermann:
\begin{equation}
\lambda_i(d_m, \omega) \approx \sqrt{14 + 2 \cdot\log \frac{\omega\cdot d_m}{2} - 2 \cdot \log(.007)} \cdot 0.02
\end{equation}
- Harris T A and Kotzalas M N (2007) Rolling Bearing Analysis - Essential Concepts of Bearing Technology. CRC Press, Taylor and Francis Group
- Palmgren A (1959) Ball and Roller Bearing Engineering. 3rd ed. Burbank, Philadelphia
- International Organization for Standardization (ISO) (2007) ISO DIN 281:2007 Rolling bearings - Dynamic load ratings and rating life. Beuth Verlag: Berlin
- Weidermann, F. (2001) ‘Praxisnahe thermische simulation von Lagern und Führungen in Werkzeugmaschinen’, CAD-FEM User’s Meeting 19. Potsdam.