# Energy-Based Prediction of the Displacement of DCFP Bearings
## 6 Proposed Prediction Method
### Mechanical Model
#### 6.1.2 Prediction of the Response Displacement of the Isolation Layer  
In this section, the peak response displacement changes in the isolation layer, \(\Delta y_{i}\), were primarily predicted by simplifying Figure 14. It should be mentioned that \(\Delta y_{i}\) was predicted instead of \(y_{max}\). This was because the value of \(y_{max}\) was highly related to former response displacement, which made it difficult to be derived theoretically. However, the derivation of \(\Delta y_{i}\) was more related to the corresponding velocity pulse. Furthermore, since the maximum \(\Delta y_{i}\) was usually larger than \(y_{max}\), the maximum \(\Delta y_{i}\) could be considered to be a conservative value of \(y_{max}\). As shown in the left figure of Figure 16, the solid line shows a ground velocity pulse from \(vg_{i-1}\) to \(vg_{i}\) to \(vg_{i+1}\) and the two dashed lines show the slope from \(vg_{i-1}\) to \(vg_{i}\) and the slope from \(vg_{i}\) to \(vg_{i+1}\), respectively, which can be understood as the average ground acceleration. The dotted line is the absolute velocity of the upper structure, and its slope can be roughly considered to be constant, as introduced in Figure 14. Furthermore, since the points where \(vu\) and \(vg\) intersect are determined by the former velocity history, the intersection in the left figure of Figure 16 was randomly picked between \(vg_{i-1}\) and \(vg_{i}\), referred to as \(vu_{i-1}\). At the intersection points, vu was equal to vg, which meant that \(v\) was equal to zero, and this meant that the direction of the response displacement of the isolation layer, \(y\), would change from the next second. Then, until the next intersection, \(y\) would be in the same direction; reach the peak displacement change, \(\Delta y_{i}\); and reverse direction again. Then, how the calculation of \(\Delta y_{i}\) can be simplified under a large velocity pulse is shown in the right figure of Figure 16. In the simplification, \(vu_{i-1}\) and \(vg_{i-1}\) are assumed to be equal, which is a conservative assumption for predicting \(\Delta y_{i}\):  
\[vu_{i-1}=vg_{i-1} \tag{17}\]  
Figure 16: Method to predict the response displacement of the isolation layer (\(0<ku_{i}/kg_{i}<0.8\)). Figure 15: Time history of the relative velocity between the upper and lower concave plates during earthquakes where the NCC with PGV are equal to 25 mm/s. The acceleration between two adjacent peak ground velocities is assumed to be the average value, \(kg_{i}\) and \(kg_{i+1}\):  
\[kg_{i}=\frac{vg_{i-vg_{i-1}}}{\Delta t_{i}};\ \ \ kg_{i+1}=\frac{vg_{i+1}-vg_{i}}{ \Delta t_{i+1}} \tag{18}\]  
where \(\Delta t_{i}\) is the time interval between \(vg_{i-1}\) and \(vg_{i}\); \(\Delta t_{i+1}\) is the time interval between \(vg_{i}\) and \(vg_{i+1}\). The friction coefficient was assumed to be the nominal friction coefficient, \(\mu_{n}\), which was equal to 0.043 and could be used to roughly simulate the behavior of the bearing under earthquake excitations. Therefore,  
\[ku_{i}=\mu_{n}\times g\times kg_{i}/\left|kg_{i}\right| \tag{19}\]  
where \(g\) is the gravity acceleration in mm/s\({}^{2}\). Then, the predicted value will be \(\Delta y_{i}\)_-pre_, as shown in Figure 16, which is the predicted peak response displacement of the isolation layer. It can be seen from Figure 16 that, when the value of \(ku_{i}\) is close to \(kg_{i}\), \(\Delta y_{i}\)_-pre_ is close to zero. However, based on the response analysis results, \(\Delta y_{i}\) can also be relatively large in this situation. Therefore, a special prediction method was proposed for these cases, as shown in Figure 17, in which the case of \(ku_{i}\)/\(kg_{i}\) being equal to 1.0 is shown.