## 7 Pounding Tuned Mass Damper
### Modeling of a Building Equipped with Pendulum PTMD  
A pendulum PTMD is installed on the top floor of a multi-story timber building, as shown in Figure 14. The dynamic equation of motion of the building equipped with pendulum PTMD under external excitation is developed using the Lagrange method. Theoretical derivation and extensive numerical study of new pendulum PTMD are available in [PERSON] and [PERSON]'s study [85]. This study modified the equations of single-degree-of-freedom systems for multi-degree-of-freedom systems. The equation of motion of a building with a pendulum PTMD under earthquake excitation is given by:  
\[\left(\mathbf{M}_{\mathrm{s}}+\mathbf{L}_{\mathrm{d}}m\mathbf{L}_{\mathrm{d}} ^{\mathrm{T}}\right)\ddot{\mathbf{X}}_{\mathrm{s}}+\mathbf{L}_{\mathrm{d}}m \mathbf{L}_{\mathrm{d}}^{\mathrm{T}}l\ddot{\theta}-\mathbf{L}_{\mathrm{d}}m \mathbf{L}_{\mathrm{d}}^{\mathrm{T}}l\ddot{\theta}^{2}\theta+\mathbf{C}_{ \mathrm{s}}\dot{\mathbf{X}}_{\mathrm{s}}+\mathbf{K}_{\mathrm{s}}\mathbf{X}_{ \mathrm{s}}=-\mathbf{M}_{\mathrm{s}}\Lambda\ddot{x}_{g}-\mathbf{L}_{\mathrm{d} }P\ \right\} \tag{10}\]  
\[I=I_{cm}+ml^{2} \tag{11}\]  
where \(\mathbf{M}_{\mathrm{s}}\), \(\mathbf{K}_{\mathrm{s}}\), and \(\mathbf{C}_{\mathrm{s}}\) are the building's mass, stiffness, and damping, as previously defined. \(m\) is the pendulum mass equal to the equivalent mass of TMD defined in Section 4, \(l\) is the pendulum length, \(\theta\) is the angular displacement of the PTMD, \(\mathbf{L}_{\mathrm{d}}\) is the vector containing the damper location, \(I_{cm}\) is the moment of inertia of the PTMD, \(g=9.8\) m/s\({}^{2}\), and \(P\) is the bounding force. Recently, there has been intense research to model the impact force between adjacent tall buildings under seismic events [86; 87; 88]. The linear spring model was used initially to represent the impact force but fails to account for energy dissipation during collision [89]. The Kelvin impact model combines linear spring with a damper, where the damping coefficient is determined to reflect energy dissipation. However, the Kelvin model could not represent the bounding as a highly nonlinear phenomenon. Therefore, an impact model based on Hertz contact law has been suggested lately to include the nonlinear nature of the bounding force [90]. Based on the Hertz law, the bounding force at any given time can be expressed as:  
\[P=\left\{\begin{array}{l}\alpha\delta(t)^{\frac{3}{2}}+c(t)\dot{\delta}(t) \ for\ \ \delta(t)>0\ \ and,\ \ \ \ \ \dot{\delta}\left(t\right)>0\ (Approaching\ period)\\ \alpha\delta(t)^{\frac{3}{2}}\ \ \ \ \ \ for\ \ \delta(t)>0\ \ and,\ \ \ \ \ \dot{\delta}\left(t\right)<0\ (Restitution\ period)\\ 0\ \ for\ \ \ \delta(t)\ \where \(\delta(t)\) is the relative deformation, \(\dot{\delta}\left(t\right)\) is the impact velocity, \(\alpha\) is the bounding stiffness, and \(c(t)\) is the damping coefficient resulting from impact.  
In the primary impact model, a gap exists between the bounding mass and the viscoelastic boundary [90]. As the viscoelastic material is nonlinear, damping induced by an impact is not uniform; rather, it depends on the structural response. The instantaneous impact damping coefficient is defined as:  
\[c(t)=2\xi\sqrt{\alpha\sqrt{\delta(t)}\frac{Mm}{M+m}} \tag{13}\]  
\[\xi=\frac{9\sqrt{5}}{2}\frac{1-e^{2}}{e(e(9\pi-16)+16)} \tag{14}\]  
where \(M\) and \(m\) are the masses of impacting bodies. In this paper, \(M\) is considered the first modal mass of the building, and \(m\) is the mass of the damper. \(\xi\) is the damping ratio  
Figure 14: Schematic representation of a high-rise building equipped with a pendulum PTMD. developed from the impact, which relies on the coefficient of restitution \(e\)[91].