## 4 Experimental Results
### Earthquake Excitation Case
#### 4.4.2 Validation of Numerical Model  
As the proposed SPPTD constitutes a nonlinear control device, its mathematical modeling poses challenges for general-purpose FEM software. Consequently, a MATLAB R2013a-based simulation framework was developed to numerically analyze the coupled dynamics of the controlled structure-SPPTD system. Numerical model validation was conducted through comparisons between experimental data and numerical results.  
As summarized in Table 4, maximum displacement discrepancies between experimental and numerical results remain within acceptable margins: error ranges for uncontrolled and SPPTD-controlled cases were 1.00-5.36% (min-max) and 2.07-9.99% (min-max), respectively. Furthermore, Figures 9-12 demonstrate a close agreement between shake-table testing and numerical simulation results regarding the displacement responses of the controlled model. The numerical results consistently exhibit minor overestimations compared to the experimental data, and this characteristic enhances the damper's operational reliability in practical applications. These results confirm that the proposed impact force model and numerical simulation framework accurately predict structural dynamic responses under SPPTMD control.  
pronounced vibration-reduction effects on the structure regardless of variations in external excitation frequencies while exhibiting robust performance across all tested operational conditions;
3. The experimental data demonstrate that the SPPTMD effectively mitigates structural displacement responses under seismic loading, achieving a peak vibration-reduction rate of 60.25% and an RMS reduction rate of 58.35%, with all results indicating that the vibration control capability of the SPPTMD increases when the ground is softer;
4. A comparative analysis of displacement response data obtained from shake-table testing and numerical simulations revealed close agreement in displacement time-history curves, demonstrating that the implemented impact force model and equations of motion accurately capture the dynamic behavior of the SPPTMD-structure coupled system.  
In this study, the SPPTMD was demonstrated to effectively reduce the dynamic response of structures under external dynamic excitations (e.g., wind or seismic loads) while exhibiting excellent robustness. Given these demonstrated capabilities, the SPPTMD shows strong potential for application in vibration control of tall and slender structures, including high-rise buildings, observation towers, television towers, and transmission towers. This study provides a comprehensive theoretical and experimental foundation to support future academic research in this field.  
Conceptualization, Q.W., L.Z., and H.L.; methodology, Q.W. and H.L.; software, Q.W. and X.G.; validation, Q.W. and H.L.; data curation, H.L. and Q.W.; writing--original draft preparation, Q.W.; writing--review and editing, H.L., L.Z., and Q.W.; funding acquisition, H.L. All authors have read and agreed to the published version of the manuscript.  
The authors would like to express our sincere gratitude for the financial support of the National Major Scientific Instruments and Equipments Development Project of National Natural Science Foundation of China (52027811) and the National Natural Science Foundation of China (51738007).  
Not applicable.  
The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
Authors [PERSON] and [PERSON] were employed by the company Liaoning Urban and Rural Construction Planning Design Institute Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.