## 4 Optimal Design of Passive Energy Dissipation Systems
### Tuned Mass Damper  
TMD is a motion-based passive system that consists of a mass \(m_{d}\), a spring with spring stiffness \(k_{d}\), and a dashpot with a damping coefficient \(c_{d}\) attached and typically tuned to the natural structural frequency [34]. During an earthquake, the damping system is stretched and compressed, reducing vibrations in the structure by increasing its effective damping [34]. A schematic representation of the 2 DOF (degree of freedom) system is shown in Figure 3, noting that \(m,~{}k\), and \(c\) represent, respectively, the main mass, stiffness, and inherent damping coefficient of the structural system. TMD is typically effective over a narrow frequency band. It is therefore important to be tuned to a particular natural frequency. The system efficiency decreases, with structures having several closely spaced natural frequencies [34]. An appreciation of TMD efficiency can be obtained by following the basic development of Den Hartog [6], which considers an undamped structural system subject to a sinusoidal excitation. Figure 4 shows that the dynamic amplification factor, \(R\), which takes the  
Figure 3: Schematic representation of the 2 DOF system. Figure 2: (**a**) The bidirectional recorded accelerograms, (**b**) response spectra relative to the selected accelerograms compared to the EC8 elastic response spectrum Type 1 with peak ground acceleration equal to 0.3 g, ground type B, and 5% damping. damping effect of the TMD, is a function of the four essential variables: the mass ratio \(\overline{m}\) (\(\overline{m}=m_{d}/m\)), the TMD damping ratio \(\xi_{d}\) (\(\xi_{d}=\frac{\xi_{d}}{2\cdot m_{d}\cdot\omega_{d}}\)), the frequency ratio \(\
u\) (\(\
u=\omega_{d}/\omega\)), and the forced frequency ratio \(\lambda\) (\(\lambda=\overline{\omega}/\omega\)), where \(\omega=\sqrt{k/m}\) and \(\omega_{d}=\sqrt{k_{d}/m_{d}}\) are the natural frequency of the structural system and TMD, respectively. The dynamic amplification factor \(R\) is expressed by the Equation (1):  
\[R=\sqrt{\frac{\left(\
u^{2}-\lambda^{2}\right)^{2}+\left(2\cdot\xi_{d}\cdot \
u\cdot\lambda\right)^{2}}{\left[(\
u^{2}-\lambda^{2})(1-\lambda^{2})-\
u^{2 }\cdot\lambda^{2}\cdot\overline{m}\right]^{2}+\left(2\cdot\xi_{d}\cdot\
u\cdot \lambda\right)^{2}(1-\lambda^{2}-\lambda^{2}\cdot\overline{m})^{2}}} \tag{1}\]  
Figure 4 shows a plot of \(R\) as a function of \(\lambda\) for \(\overline{m}=0.05\) and \(\
u=1\). Without TMD damping, the response amplitude is infinite at two resonant frequencies of the 2 DOF systems. Furthermore, for an infinite TMD damping, the two masses are virtually fused to each other, leading the amplitude of resonant frequency to be infinite again [1]. Therefore, between these extremes, there is a value of \(\xi_{d}\), for which the peak becomes a minimum. An objective of installing TMD in structures is to bring the response amplitude down to its lowest possible value, 1; this is why the damping ratio of TMD must be carefully selected in such a way that small amplifications over a wider frequency bandwidth can be achieved. As can be seen in Figure 4, this can be achieved by taking a small value of \(\xi_{d}\), like 30%. So, the effect of the TMD damping ratio is very essential. One observes that this parameter must exist but must not be high, because at this case, the amplifications are small and the frequency range in which the damper works is the biggest increasing damper efficiency. Outside of this range, the motion is not considerably influenced by the TMD system. The conventional TMD described above requires a large mass and space for installation, thus creating architectural constraints [35]. An alternative approach is using a pendulum configuration PTMD. During ground motion, the pendulum produces a horizontal force which opposes the story motion [36].