# Energy-Based Prediction of the Displacement of DCFP Bearings
## 4 Numerical Simulation of the Experiments  
### Friction Dependency  
The pressure dependency of the friction coefficient could be considered by a pressure dependency factor \(\gamma\), which is related to the bearing stress \(\sigma\) at the contact area of the concave plate and the slider. The pressure dependency equation was obtained by a previous experimental study [14]:  
\[\gamma=2.03\times\sigma^{-0.19}+0.068 \tag{7}\]  
The dependency of the friction coefficient on velocity was considered by a velocity dependency factor \(\alpha\). This factor was related to the velocity of the upper concave plate relative to the lower concave plate \(v\), which could be described by the following equation [14]:  
\[\alpha=1-0.55\times e^{-0.019v} \tag{8}\]  
Figure 3 shows the applicability verification of the pressure and velocity dependency equations for \(\varphi\)300 and \(\varphi\)400 specimens, based on the dependency test introduced in Table 1. In Figure 3, the experimental friction coefficients calculated from the dependency test were normalized by the \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Spec.** & **Test** & **Pressure** & **Amplitude** & **Vmax** & **Period** & **Cycle** \\ \hline
**num.** & **num.** & **N/mm2** & \(\pm\)**mm** & **mm/s** & **s** & **num.** \\ \hline \multirow{4}{*}{\(\varphi\)300 \(\varphi\)400} & T01 & 60 & 200 & 400 & 3.14 & 4 \\ \cline{2-6}  & \multicolumn{6}{c}{(2) Pressure dependency test} \\ \cline{2-6}  & T02 & 40 & 20 & 62.83 & 4 \\ \cline{2-6}  & T03 & 60 & 200 & 20 & 62.83 & 4 \\ \cline{2-6}  & T04 & 80 & 20 & 62.83 & 4 \\ \hline \end{tabular}
\end{table} 
Table 1: Test procedure of the dependency test. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Spec.** & **Test** & **Pressure** & **Amplitude** & **Vmax** & **Period** & **Cycle** \\ \hline
**num.** & **num.** & **N**\(\backslash\)**mm2** & \(\pm\)**mm** & **mm/s** & **s** & **num.** \\ \hline \multirow{4}{*}{\(\varphi\)300 \(\varphi\)400} & T01 & 60 & 268 & 392 & 4.26 & 3 \\  & T02 & & 10 & 14.646 & 4.26 & 20 \\  & T03 & & 100 & 146.369 & 4.26 & 3 \\  & T04 & 60 & 200 & 292.738 & 4.26 & 3 \\  & T05 & & 268 & 392.269 & 4.26 & 3 \\  & T06 & & 400 & 585.476 & 4.26 & 3 \\  & T07 & & 400 & 585.476 & 4.26 & 3 \\ \(\varphi\)300 \(\varphi\)400 & T08 & 40 & 400 & 585.476 & 4.26 & 3 \\  & T09 & 80 & 400 & 585.476 & 4.26 & 3 \\  & T10 & 30 & 440 & 644.024 & 4.26 & 1 \\  & T11 & 90 & 440 & 644.024 & 4.26 & 1 \\  & T12a & & & & & 7 \\  & T12b & & 300 & 439.107 & 4.26 & 7 \\  & T12c & & & & & 6 \\  & T13 & & 268 & 392.269 & 4.26 & 3 \\ \hline \end{tabular}
\end{table} 
Table 2: Test procedure of the ASCE test. values at 60 N/mm\({}^{2}\) and 400 mm/s, respectively, in order to minimize the influence of product variation. As a result, Equations (7) and (8) are both highly consistent with the experimental results, and thus, the previously proposed pressure and velocity dependency equations are effective for DCFP bearings with \(\varphi\)300 and \(\varphi\)400 slider diameters. The temperature dependency equation is as follows:  
\[\beta=1.13\times\exp(-0.007T) \tag{9}\]  
where \(\beta\) is the temperature dependency factor and \(T\) is the temperature of the contact area between the slider and the concave plate in Celsius [15; 16]. As the temperature, \(T\), is difficult to accurately measure during testing, a numerical method proposed by [PERSON], [PERSON] et al. [4; 17] will be introduced to simulate the temperature in the next section. The friction coefficient at each time during excitation can be calculated by:  
\[\mu=\mu_{0}\times\gamma\times\alpha\times\beta \tag{10}\]  
where \(\mu\), \(\gamma\), \(\alpha\) and \(\beta\) are the friction coefficient, pressure dependency factor, velocity dependency factor and temperature dependency factor at each time, respectively, and \(\mu_{o}\) is the friction coefficient at 60 N/mm\({}^{2}\) (\(\gamma\) = 1), 400 mm/s (\(\alpha\) = 1) and an atmosphere temperature of around 20 \({}^{\circ}\)C (\(\beta\) = 1), which was selected as 0.075 for all the specimens in this study.