## 2 Mechanism of SPPTMD and Numerical Model of Structure-SPPTMD System
### Numerical Model of the Structure-SPPTMD System  
Assuming that an SPPTMD is attached to the \(i\)th degree of freedom (DOF) of an n-DOF system (Figure 3 [51]), the equation of motion of the structure-SPPTMD system is as follows:  
\[\left\{ \begin{array}{l}\mathbf{M_{s}}\ddot{x}(t)+\mathbf{C_{s}}\dot{x}(t)+ \mathbf{K_{s}}x(t)=-\mathbf{M_{s}}\ddot{x}_{g}(t)+\mathbf{L}F_{\mathrm{s,x}}(t )+F_{dir}\mathbf{L}F_{c,x}(t)\\ m_{d}\ddot{x}_{d}(t)=-m_{d}\ddot{x}_{g}(t)-F_{\mathrm{s,x}}(t)-F_{dir}F_{c,x}(t )\\ m_{d}\ddot{y}_{d}(t)=m_{d}g-F_{g,g}(t)-F_{c,g}(t)\end{array} \right., \tag{4}\]  
where \(\mathbf{M_{s}}\), \(\mathbf{C_{s}}\), and \(\mathbf{K_{s}}\) are the mass matrix, damping matrix, and stiffness matrix of the uncontrolled structure, respectively; \(\ddot{\mathbf{x}}(t)\), \(\dot{\mathbf{x}}(t)\), and \(\mathbf{x}(t)\) are the acceleration vector, velocity vector, and displacement vector of the uncontrolled structure, respectively; \(\mathbf{I}\) is a column vector of \(1\)s; \(\ddot{x}_{g}\) is the ground acceleration; \(\mathbf{L}\) is the position vector of the SPPTMD; \(F_{\mathrm{s,x}}\) and \(F_{c,x}\) are the restoring force and the bounding force, respectively, of the additional mass in the x-direction; \(F_{dir}\) is the direction of the bounding force; \(m_{d}\) is the additional mass; \(x_{d}\) and \(y_{d}\) are the displacements of the mass block in the horizontal and vertical directions, respectively; \(F_{\mathrm{s,y}}\) and \(F_{c,y}\) are the restoring force and bounding force, respectively, of the additional mass in the y-direction. The position vector \(\mathbf{L}\) of the SPPTMD can be written as follows:  
\[\mathbf{L}=\left(0,0,\cdots,\underbrace{1}_{\text{the $i$th element equals one}},0,\cdots,0\right)^{T}, \tag{5}\]  
The circular shape of the limiter allows the bounding to occur on both the left and right sides. The bounding force direction \(F_{dir}\) is expressed as follows:  
\[F_{dir}=\left\{ \begin{array}{l l}1&x_{d}-x_{i}>g_{p}&\text{ bounding on the right side}\\ -1&x_{d}-x_{i}<-g_{p}&\text{ bounding on the left side}\\ 0&\text{others}&\text{not bounding}\end{array} \right., \tag{6}\]  
Figure 2: Schematic of the SPPTMD. According to our previous study, the restoring force and bounding force are obtained as follows:  
\[\left\{ \begin{array}{l}F_{s,x}=k_{s}x_{r}(1-\frac{l_{ori}}{\sqrt{x_{r}^{2}+(y_{ q}+l_{o})^{2}}})\\ F_{s,y}=k_{s}(y_{d}+l_{o})(1-\frac{l_{ori}}{\sqrt{x_{r}^{2}+(y_{d}+l_{o})^{2}}}) \end{array} \right., \tag{7}\]  
\[x_{r}=x_{d}-x_{i}, \tag{8}\]  
\[l_{ori}=l_{o}-\frac{m_{d}g}{k_{s}}, \tag{9}\]  
where \(k_{s}\) is the stiffness of the spring, \(x_{i}\) is the horizontal displacement of the \(i\)th DOF of the uncontrolled structure, and \(l_{ori}\) is the original length of the spring. According to the Hertz contact force model, the bounding force of the SPPTMD is obtained as follows:  
\[F_{c}=\left\{\begin{array}{l}\beta\delta^{1.5}+c\delta\ \ \text{if}\ \dot{\delta}>0\\ \beta\delta^{1.5}\  
data. The SPPTMD designed for vibration suppression in the model features a mass ratio of 3.4%, a spring stiffness of 80.12 N/m, and a pendulum length of 0.11 m, as determined through numerical calculations. The impact stiffness of the SPPTMD in this study adopts the experimentally validated value \(k_{h}=1.7\times 10^{4}\)N/m\({}^{3/2}\), derived from [PERSON] et al. 's investigations [25] on viscoelastic materials, thereby ensuring consistency between numerical modeling and empirical data while maintaining parametric fidelity. The motion limiter consisted of a polyvinyl chloride (PVC) pipe internally clad with viscoelastic damping material, maintaining a 7.5 mm operational gap between the auxiliary mass and limiter surfaces (Figure 4). The SPPTMD assembly was rigidly affixed to the upper floor slab via bolted connections. The sinusoidal excitation and historical earthquake excitation were administered using a shaking table (Wavespectrum Century Technology, Beijing, China).