# Energy-Based Prediction of the Displacement of DCFP Bearings
## 6 Proposed Prediction Method
### Mechanical Model  
1.1 Relationship between Ground Velocity, the Response Displacement and the Input Energy of the Isolation Layer. By using the model in Figure 9, the relative movement and the energy balance in a base-isolated building become the relative movement and the energy balance in the base-isolated layer. As one matter of concern during the design stage, the prediction of the maximum relative displacement between the upper and lower concave plates of the bearing during earthquakes is the focus of this study. Figure 12 shows the relative movement of the base-isolated layer during strong earthquake pulses, where \(\upsilon u\) is the absolute velocity of the upper structure, \(\upsilon g\) is the absolute velocity of the ground, and \(y\) shows the relative displacement between the upper and lower concave plates, which can also be called the response displacement of the isolation layer. Then, the response velocity of the isolation layer (the relative velocity between the upper and lower concave plates) can be expressed as:  
\[v=\dot{y}=\upsilon u-\upsilon g \tag{13}\]  
During a strong earthquake pulse, the history of \(\upsilon g\) is known and the value of \(\upsilon u\) is mainly determined by the dynamic friction coefficient of the bearing. Therefore, \(y\) can be predicted by integral \(v\), which is the difference between \(\upsilon u\) and \(\upsilon g\). Figure 11: Relationship between the maximum response displacement of the isolation layer and the peak ground velocity under unidirectional earthquake inputs. Based on the response analysis results, the ground velocity record of the earthquake was found to be greatly related to, and work simultaneously with, the transferred energy and the response displacement of the isolation layer during earthquakes. As shown in Figure 13, during one earthquake, several large velocity pulses (from one ground peak velocity, \(vg_{i-1}\), to the next peak, \(vg_{i}\), and to the next peak, \(vg_{i+1}\), is one velocity pulse) exist. The difference between \(vg_{i-1}\) and \(vg_{i}\) was defined as the velocity difference of this velocity pulse, \(\Delta vg_{i}\), which could show the intensity of the velocity pulse:  
\[\Delta vg_{i}=vg_{i}-vg_{i-1} \tag{14}\]  
Figure 12: Relative movement of the base-isolated layer. Figure 13: Time history of ground velocity, input energy and response displacement of the base-isolated layer during NCC (Northridge earthquake measured by Canyon Country-WLC station) with its PGV equals to 250 mm/s. It was found that, when a large \(\Delta vg_{i}\) exists in the earthquake, it will lead to large input energy in the structural system, \(\Delta E_{i}\), and a large response displacement of the isolation layer (a large relative displacement between the upper concave plate and the lower concave plate), \(\Delta y_{i}\), simultaneously. Since the upper structure is considered to be a rigid body in this study, the total input energy, \(E\), is consumed by the hysteresis energy \(E_{h}\). However, during a velocity pulse, some of the input energy will first be temporarily stored in the kinetic energy, \(E_{k}\), and the elastic strain energy, \(E_{s}\). In order to eliminate the interference of \(E_{k}\), the change in energy from the beginning of the velocity pulse to the time kinetic energy recovers to zero, which is approximately the time interval from \(y_{i-1}\) to \(y_{i}\) and is considered to be very useful as it is related to the peak response displacement change, \(\Delta y_{i}\). In Figure 13, \(y_{i}\) and \(y_{i-1}\) are two neighboring peak response displacements of the isolation layer, and every time a peak response displacement is achieved, \(E_{k}\) becomes zero; \(y_{max}\) is the maximum amplitude of the response displacement of the isolation layer; \(\Delta y_{i}\) is one of the peak response displacement changes of the isolation layer, and the one shown in the figure is the maximum. Then, in this time interval, the total input energy, \(\Delta Er_{i}\), can be expressed as the sum of the consumed energy.