## 3 Methodology
### System Parameters and Response Quantities Definitions
#### 3.2.2 Ratio of Uncoupled Vibration Frequencies, \(\Omega\)  
Different values of the uncoupled frequency ratio \(\Omega\) are obtained by altering the location of the main lateral resisting structural elements from the perimeter of the building to near the core as shown in Figures 2-4. For symmetric buildings under study, the fundamental torsional and lateral frequencies \(\omega_{0}\), \(\omega_{x}\) and \(\omega_{y}\) depend on the torsional and lateral stiffness matrices \(K_{0}\), \(K_{x}\), and \(Ky\), respectively. These frequencies can be computed using Eigen value or Ritz methods of structural dynamics. However, for unsymmetrical buildings the standard procedures are not applicable as the results will show coupled translational-torsional motion in each mode. [PERSON] et al. [7] proposed a method to calculate the uncoupled frequencies of unsymmetrical structures.  
Figure 4: Building plans for third group of parametric study (with \(b\prime\) = 2.45): (**a**) Building frequency ratio \(\Omega\) = 1.5; (**b**) Building frequency ratio \(\Omega\) = 1.4; (**c**) Building frequency ratio \(\Omega\) = 1.3; (**d**) Building frequency ratio \(\Omega\) = 1.15; (**e**) Building frequency ratio \(\Omega\) = 1.1; (**f**) Building frequency ratio \(\Omega\) = 1.0; (**g**) Building frequency ratio \(\Omega\) = 0.85; (**h**) Building frequency ratio \(\Omega\) = 0.7.  
Figure 3: Building plans for second group of parametric study (with \(b\prime\) = 1.4): (**a**) Building frequency ratio \(\Omega\) = 1.4; (**b**) Building frequency ratio \(\Omega\) = 1.3; (**c**) Building frequency ratio \(\Omega\) = 1.2; (**d**) Building frequency ratio \(\Omega\) = 1.1; (**e**) Building frequency ratio \(\Omega\) = 1.0; (**f**) Building frequency ratio \(\Omega\) = 0.9; (**g**) Building frequency ratio \(\Omega\) = 0.8; (**h**) Building frequency ratio \(\Omega\) = 0.7.  
Figure 2: Building plans for first group of parametric study (with \(b\prime\) = 1.1): (**a**) Building frequency ratio \(\Omega\) = 1.4; (**b**) Building frequency ratio \(\Omega\) = 1.3; (**c**) Building frequency ratio \(\Omega\) = 1.2; (**d**) Building frequency ratio \(\Omega\) = 1.1; (**e**) Building frequency ratio \(\Omega\) = 1.0; (**f**) Building frequency ratio \(\Omega\) = 0.9; (**g**) Building frequency ratio \(\Omega\) = 0.8; (**h**) Building frequency ratio \(\Omega\) = 0.7.  
Once \(\omega_{y}\) and \(\omega_{\theta}\) are known, the uncoupled angular frequency ratio \(\Omega\ =^{\omega_{\theta}}/\omega_{y}\) is computed. Studies have shown that the resulting value of \(\Omega\) is more accurate than \(\omega_{y}\) and \(\omega_{\theta}\) separately, as the errors present in computation of the frequencies tend to decrease when their ratio is determined [6].  
A large value of the ratio \(\Omega\), which may be viewed as a property of the building, indicates that the building is torsionally stiff, such as the case when stiff resisting structural elements are located near the perimeter of the building plan. On the other hand, a small value of \(\Omega\) indicates the building is torsionally flexible, which is the case when stiff structural elements are located near the central core of the building but flexible elements are located at the perimeter. Accordingly, a boundary value between flexible and stiff torsional systems is defined at \(\Omega\) = 1. From practical point of view, most buildings have value of \(\Omega\) ranging between 0.7 and 1.5 [8].