# Enhancing Friction Pendulum Isolation Systems Using Passive and Semi-Active Dampers
## 3 Isolator Design
### Optimization Problem  
A two-step design process based on minimizing two performance functions \(\phi_{1}\) and \(\phi_{2}\) is proposed. The objective is to design, firstly the friction coefficient \(\mu\) and secondly, the plate size (the pendulum natural frequency \(f_{a}\)) and the normal functioning damping coefficient \(c_{a}\) (for the passive version), or \(c_{min}\) (for the semi-active version). The first step consists of minimizing a performance function \(\phi_{1}\) and obtaining the optimum friction coefficient \(\mu\) for the single FPS (without damper) under serviceability condition is such a way that good re-centering capacity is assured. In a second step, under the design earthquake, the frequency of the pendulum and the normal function damping coefficient are obtained by minimizing a performance function \(\phi_{2}\). Thus, the two-step design process is as follows:  
* Step 1. Single FPS (Figure 4a) under the serviceability earthquake. The performance function \(\phi_{1}\) is defined as: \[\phi_{1}(z,\mu,f_{a})=a_{1}J_{1,P}+a_{2}J_{2,P},\] (13) in which \(z=[m_{p},k_{p},c_{p},m_{a}]\), \(a_{1}=a_{2}=0.50\) are the weighting factors. The optimum value of \(\mu\) for the range of \(f_{a}\) (plate sizes) considered is calculated from the: \[\min_{\mu}\quad\phi_{1}(z,\mu,f_{a})\] (14) subject to \[f_{a}(z)\in[f_{a,min};f_{a,max}]\,,\forall_{\mu}\in[\mu_{min};\mu_{max}]\,.\] (15)
* Step 2. FPS+VD or FPS+SD (Figure 4b,c) under the design earthquake. The performance function \(\phi_{2}\) is defined as: \[\phi_{2}(z,c_{min},f_{a})=\sum_{i=1}^{3}a_{i}j_{i,P}+\sum_{i=1}^{3}b_{i}j_{i,RMS},\] (16) with \(\sum_{i=1}^{3}(a_{i}+b_{i})=1\) and \(z=[m_{p},k_{p},c_{p},m_{a},\mu,c_{max}]\), where \(a_{i}\) and \(b_{i}\) are the weighting factors of control effects adjustment for a balance between the peak and RMS PIs, respectively. Generally speaking, \(\phi_{2}\) is a performance function that considers a balance between structural magnitudes (\(J_{1}\) and \(J_{3}\)), acceleration and inter-story drift, and the plate displacement \(J_{2}\), since this magnitude is normalized to the minimum plate size displacement. The minimization of \(\phi_{2}\) will reduce simultaneously, the structural magnitudes and the plate size. In this sense, the continuous movement of the bearing plate (\(J_{2,RMS}\)) should not be penalized (\(b_{2}=0\)), and however, the peak value of the plate displacement (\(J_{2,P}\)) should be penalized (\(a_{2}=1/6\)) in order to choose the minimum plate size that provides good performance (as it was shown in Figures 10 and 11).  
Figure 12: \(J_{3,RMS}\) for energy-controlled FPS+SD under the design earthquake.  
Regarding the structure performance, \(\phi_{2}\) pays attention mainly to the peak of the inter-story drift \(J_{3,P}\) (\(a_{3}=1/3\)) since this magnitude is crucial to avoid structural damages during the event. The remainder of the PIs, peak and RMS structure acceleration (\(J_{1,P}\), and \(J_{1,RMS}\)) and the RMS of the inter-story drift are equally penalized (\(1/6\)) with a weight factor which is half of the peak of the inter-story drift. Thus, the optimum damping coefficient \(c_{a}\) for FPS+VD or \(c_{min}\) for FPS+SD (when the MR damper is normally functioning), and the pendulum natural frequency \(f_{a}\)(which defines the plate size) are calculated from the minimization of \(\phi_{2}\) as follows:  
\[\min_{c_{min},f_{a}}\quad\phi_{2}(\underline{z},c_{min},f_{a}), \tag{17}\]  
\[\text{s\text{uject to }}f_{a}(\underline{z})\in\left[f_{a,min};f_{a,max}\right],c _{min}\in\left[0;c_{max}\right]. \tag{18}\]  
Note that \(c_{min}=c_{a}\) in Equation (17) if the FPS+VD is being designed. Note also that the proposed design process requires two sequential inputs, first, a low-intensity earthquake followed by a design earthquake corresponding to the considered seismic area. The selection of the design earthquake according to current codes has been explained in Section 3.1.