# Free Vibration Analysis of Arches with Interval-Uncertain Parameters
## 5 Conclusions  
The linear natural frequencies of arches under interval-uncertain parameters were analyzed in this study. By coupling the DQ method and the Chebyshev approximation theory, a surrogate model was established to solve the uncertainty problem with respect to the dynamic characteristics of arches. The non-intrusive interval method is valid in free-vibration analysis, for which probabilistic methods cannot be utilized due to a lack of essential prior information. The accuracy and effectiveness of the proposed CPS model were demonstrated via a comparative study of the natural frequency ranges of the arch obtained using the MCS and the model. It is clear that the CPS model is more efficient, and this will be more obvious in multiple-uncertainty situations, because the sample points in multiple dimensions are enormous, while the interpolation numbers used in the CPS model remain acceptable. Based on the results of several cases exhibited in the foregoing sections, the following conclusions can be made:  
1. Some geometrical parameters, such as the cross-sectional height, were found to be sensitive, which will cause large fluctuations in the dynamic properties.
2. The influence of most uncertain physical parameters on the natural frequencies exhibits a linear relationship which remains unaffected by variations in radius and central angle. However, different parameters yield distinct rates of change in frequency.
3. In the case of multi-dimensional uncertainty, the changing rate of frequencies surpasses that of any individual parameter, as it represents a linear superposition of the combined influence exerted by each parameter.  
The analysis method employed in this study is non-intrusive and convenient, and has been demonstrated to be an efficient approach for addressing the free-vibration problems of arches with uncertainties. It is widely accepted that, despite the uncertain parameters, there should be a linear relationship between the natural frequency of the linear elastic arch and the deviation degree of an uncertain parameter. However, this study has revealed intriguing variations in the influences of different parameters on the natural frequency, sometimes exhibiting elusive changes. The methodology employed can be extended to calculate both natural frequencies and dynamic responses for nonlinear arches in future studies, which is expected to yield more valuable insights.  
Figure 8: The changing rates of the first fourth-order natural frequencies of an arch with multiple simultaneously uncertain parameters and a radius of 3 m: (**a**) first order; (**b**) second order; (**c**) third order; and (**d**) fourth order.  
**Author Contributions:** Methodology and software, Z.N., Y.Y., C.F. and J.Z.; writing--review and editing, Z.N.; validation and formal analysis, X.R. and Y.Y.; supervision, X.R. and Y.Y.; project administration, Z.N. and C.F. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (11972295).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Data are contained within the article.  
**Conflicts of Interest:** The authors declare no conflict of interest.