Bearing
(Bearing)


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-groove00.00050.55
Angular-contact30-400.0010.33
Thrust900.00080.33
Double-row, self-aligning10 0.00030.40

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 TypeGreaseOil MistOil BathOil Jet
Cylindrical roller with cage0.82.13.13.1
Cylindrical roller full complement7.5 - 7.5 -
Thrust cylindrical roller9 - 3.58

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}

  1. Harris T A and Kotzalas M N (2007) Rolling Bearing Analysis - Essential Concepts of Bearing Technology. CRC Press, Taylor and Francis Group
  2. Palmgren A (1959) Ball and Roller Bearing Engineering. 3rd ed. Burbank, Philadelphia
  3. International Organization for Standardization (ISO) (2007) ISO DIN 281:2007 Rolling bearings - Dynamic load ratings and rating life. Beuth Verlag: Berlin
  4. Weidermann, F. (2001) ‘Praxisnahe thermische simulation von Lagern und Führungen in Werkzeugmaschinen’, CAD-FEM User’s Meeting 19. Potsdam.

Inputs

ParameterSymbolUnitDescription
ForceAxial\(F_a\)NForce acting on the bearing in axial direction
ForceRadial\(F_r\)NForce acting on the bearing in radial direction
RotSpeed\(\omega\)HzRelative rotational speed between in inner and outer ring of the bearing
Temperature1\(T_o\)KTemeprature on side 1 (outer) of the bearing
Temperature2\(T_i\)KTemperature on side 2 (inner) if the bearing

Outputs

ParameterSymbolUnitDescription
Torque\(M_b\)NmResulting firctional torque
PLoss\(\dot{Q}_b\)WResulting heat loss due to frictional losses
HeatFlux\(\dot{Q}_{1\rightarrow2}\)WHeat flux to side 1 (outer) of the bearing
HeatFlux\(\dot{Q}_{2\rightarrow1}\)WHeat flux to side 2 (inner) of the bearing

Parameters

ParameterSymbolUnitDescription
NumRollingBodies\(N\)noneNumber of rolling bodies
MeanDiameter\(d_m\)mMean diameter of the bearing
LubricantMaterial\(\)Material used for the bearing lubrication
C0\(C_s\)NStatic basic load rating
ContactAngle\(\alpha_b\)degContact angle
LubricationFactor\(f0\)noneLubrication factor by Harris