## 2 The Proposed Methodology
### The Equation of Motion of Active-Controlled Structures  
Two different active-controlled structures are shown in Figure 3. In Case 1, the active control is placed only on the first floor. The structural responses such as displacement, velocity, acceleration, and total acceleration on the first floor are measured with a sensor according to the desired feedback to calculate the control signal and the corresponding control force which is applied to the first floor. In Case 2, the reactions of the control forces occur in the lower floor of the control force applied floor. In that case, the control force isreduced with the force of the upper story for this floor. For that reason, the control forces must be increased for the lower floors to provide an effective control force that is reduced by the opposite control force. In the present study, the control signal is calculated according to the responses of the first floor as in the first case and these forces must be factored with (n + i + 1) for the ith story of the n-storied structure to provide control forces that are evenly distributed to all floors as the same value. For that reason, the active control system must be applied for all stories or all control forces must be supported by the ground. In the literature, multi-story control system orientations that are supported by the ground also exist, but it is not a feasible case for a high structure since the horizontal control force will reduce while the vertical component of the force is increasing, which needs the direct increase in prestress force on the cable [7]. The equation of motion of the active tendon-controlled structures (Case 1 and Case 2) is given in Equation 10. Here, [M] is the mass matrix, [C] is the damping matrix, [K] is the stiffness matrix, K\({}_{\rm c}\) is the stiffness of the tendon, \(\alpha\) is the angle of the tendon, 1 is the unit vector, b is the influence vector of the control force and a\({}_{\rm g}\)(t) is the ground acceleration. In addition, a(t), v(t) and x(t) represent the acceleration, velocity and displacement, respectively. These matrices and vectors are shown as Equations (11)-(19) for an n degree of freedom shear structure:  
\[[\mathbf{M}]\mathbf{a}(t)+[\mathbf{C}]\mathbf{v}(t)+[\mathbf{K}]\mathbf{x}(t )=-[\mathbf{M}]\mathbf{1}\mathrm{a}_{\rm g}-(4\mathrm{k}_{\rm c}\mathrm{cos} \alpha)\mathrm{b} \tag{10}\]  
\[[\mathbf{M}]=\left[ \begin{array}{ccccc}m_{1}&&&&\\ &m_{2}&&&&\\ &&m_{3}&&\\ &&&\cdot&&\\ &&&\cdot&&\\ &&&&\cdot&&\\ &&&&\cdot&\\ &&&&\cdot&m_{n}\end{array} \right] \tag{11}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Number**. & **Name** & **Record** & **Year** & **Component 1** & **Component 2** & **Magnitude** \\ \hline
1–2 & Northridge-01 & LA–Sepulveda VA & 1994 & 0637-270 & 0637-360 & 6.7 \\
3–4 & Longa Prieta & Bran & 1989 & BRN000 & BRN000 & 6.9 \\
5–6 & Longa Prieta & Corralinos & 1989 & CLS000 & CLS000 & 6.9 \\
7–8 & Cape Mendocino & Cape Mendocino & 1992 & CPM000 & CPM00 & 7.0 \\
9–10 & Gazli, USSR & Karayr & 1976 & G\(\lambda\)Z000 & G\(\lambda\)Z090 & 6.8 \\
11–12 & Imperial Valley-06 & Bonds Corner & 1979 & H-GRK140 & H-GRK230 & 6.5 \\
13–14 & Imperial Valley-06 & Chluhuha & 1979 & H-GR1012 & H-CH1282 & 6.5 \\
15–16 & Denali, Alaska & TAPS Pune Sta. 610 & 2002 & PIS01007 & PIS0137 & 7.9 \\
17–18 & Nahanni, Canada & Site 1 & 1985 & S1010 & S1280 & 6.8 \\
19–20 & Nahanni, Canada & Site 2 & 1985 & S2240 & S2330 & 6.8 \\
21–22 & Northridge-01 & Northridge–Satiotoy & 1994 & STC0500 & STC180 & 6.7 \\
23–24 & Chi-Chi, Taiwan & TCU067 & 1999 & TCU067-E & TCU08+N & 7.6 \\
25–26 & Chi-Chi, Taiwan & TCU084 & 1999 & TCU084+E & TCU08+N & 7.6 \\
27–28 & Kocaeli, Turkey & Yarinica & 1999 & YDT060 & YDT330 & 7.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The near-fault ground motions without a pulse.