# PTMD Control on a Benchmark TV Tower under Earthquake and Wind Load Excitations
## 4 Improved Bounding Model  
### Improved Hertz Contact-Based Bounding Model  
One of the most popular contact force models used in the impact events is proposed by [PERSON] and [PERSON] and further improved for higher-impact velocities [35; 36], this model gives an expression for the hysteresis damping factor relating the kinetic energy loss due to internal damping, and it has been utilized in many domains [37; 38; 39]. More recently, [PERSON] et al. proposed a three-dimensional contact force model that can be used to simulate fully elastic to completely plastic impacts [40]. Another Hertz contact theory-based contact force model recently published was developed by [PERSON] et al. [41]. This model uses a hysteresis damping parameter that accommodates the loss of energy during the contact process. Most of these bounding models have their respective suitable domains of applications, and most of them can only be accurate when the impact objects are considered to be rigid, the contact area is small, and the impact velocity is not too large. For the particular bounding cases in the proposed PTMD, the bounding happens between a rigid mass and a VE layer with high nonlinearity. Viscoelastic material is featured by the large deformation ability and energy-dissipating capacity and, thus, energy can be further dissipated by the damping effect. A numerical bounding force model is required to decide the interaction forces between the host structure and the added mass. A non-linear viscoelastic model based on the Hertz contact law in conjunction with a damper that is active only during the approach period of the impact has been used to analyze the structural pounding in the previous study [42]. The model can be denoted as:  
\[F=\left\{ \begin{array}{ll}&\beta(u_{1}-u_{2}-g_{p})^{3/2}+c(\dot{u}_{1}-\dot{ u}_{2})&u_{1}-u_{2}-g_{p}>0\;\textit{and}\;\dot{u}_{1}-\dot{u}_{2}>0\\ &\beta(u_{1}-u_{2}-g_{p})^{3/2}&u_{1}-u_{2}-g_{p}>0\;\textit{and}\;\dot{u}_{1}- \dot{u}_{2}<0\\ &0&u_{1}-u_{2}-g_{p}<0\end{array} \right. \tag{9}\]  
where \(u_{1}\) and \(u_{2}\) are the displacements of the device and pounding layers, and \(g_{p}\) is the distance between the device and pounding layers. \(u_{1}-u_{2}-g_{p}\) is the relative pounding displacement and \(\dot{u}_{1}-\dot{u}_{2}\) is the pounding velocity. \(\beta\) is the pounding stiffness coefficient that mainly depends on material properties and the geometry of colliding bodies, and \(c\) is the impact damping which, at any instant of time, can be obtained from the formula:  
\[c=2\xi\sqrt{\beta\sqrt{u_{1}-u_{2}-g_{p}}\frac{m_{1}m_{2}}{m_{1}+m_{2}}} \tag{10}\]  
\[\xi=\frac{9\sqrt{5}}{2}\frac{1-e^{2}}{e(e(9\pi-16)+16)} \tag{11}\]  
where \(m_{1}\) and \(m_{2}\) are the masses of the two colliding bodies, and \(\xi\) is the impact damping ratio correlated with the coefficient of restitution \(e\), which is defined as the relation between the post-impact (final) relative velocity, \(\dot{u}_{1}^{f}-\dot{u}_{2}^{f}\) and the prior-impact (initial) relative velocity, \(\dot{u}_{1}^{0}-\dot{u}_{2}^{0}\), of two colliding bodies:  
\[e=\frac{\left|\dot{u}_{1}^{f}-\dot{u}_{2}^{f}\right|}{\dot{u}_{1}^{0}-\dot{u}_ {2}^{0}} \tag{12}\]  
The coefficient of restitution \(e\) can also be determined by calculating the ratio of the rebound height \(h^{\prime}\) and the original height \(h^{0}\):  
\[e=\sqrt{\frac{h^{\prime}}{h^{0}}} \tag{13}\]  
The case when \(e=1\) denotes a fully elastic collision, whereas \(e=0\) represents a perfectly plastic impact. However, the above Hertz contact-based model expressed by Equations (9)-(13) is more accurate when describing the pounding force between two layers with higher stiffness. In the previous application, this model tends to underestimate the pounding forces.