## 5 Conclusions  
This study established a bidirectional CFD-CSD coupling analysis method for blade structures, comprehensively considering the effects of aerodynamic and centrifugal loads on the structural characteristics of transonic axial flow compressor blades. It analyzed stress and deformation distributions under different conditions. Based on this, modal analysis of the blades under various prestresses was performed, and Campbell diagrams were plotted. The main conclusions are as follows:  
1. The deformation of the blades gradually increased along the blade height, with maximum deformation occurring at the lead edge tip. Regions of high stress were primarily concentrated at the upper 33% of the blade root, then radiated radially outward and gradually decreased. Both aerodynamic and centrifugal loads affected the magnitude and distribution of blade deformation and stress, but the influence of centrifugal loads on blade stress was significantly greater than that of aerodynamic loads, making centrifugal loads the primary consideration in stress calculations. 2. With increasing rotational speed, the maximum deformation of the blades showed a trend of initially increasing and subsequently decreasing, with maximum deformation under near-stall conditions exceeding that at near-choke points. However, the maximum stress of the blades exhibited a trend of initially increasing, subsequently decreasing, and then increasing, suggesting that selecting an appropriate rotational speed can reduce both the maximum deformation and maximum stress of the blades. 3. The natural frequencies of the blades increased with increasing rotational speed. Both centrifugal and aerodynamic loads had some degree of influence on the natural frequencies of the blades, but centrifugal loads were the primary factor affecting blade natural frequencies. 4. There were resonance points for compressor blades around 62% and 98% of the rotational speed. Care should be taken during compressor operation to prevent resonance, which could lead to blade damage. Conceptualization, H.L., H.B. and H.Z. ; methodology, H.L., H.B. and H.Z. ; software, H.L. ; validation, H.Z. ; formal analysis, H.L. and H.Z. ; investigation, H.L. ; resources, H.B. ; data curation, H.L. and H.Z. ; writing--original draft preparation, H.L. and H.Z. ; writing--review and editing, H.L., H.B. and H.Z. ; visualization, H.L. and H.Z. ; supervision, H.B., F.Z. and J.W. ; project \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Order** & **Mode** & **Load** & **Design Rotational** & \multicolumn{3}{c}{**Frequency (Hz)**} \\  & & & **Speed of 70\%** & **Design Rotational** & \multicolumn{1}{c}{**Design Rotational**} & \multicolumn{1}{c}{**Design Rotational**} \\  & & & **Speed of 70\%** & **Speed of 80\%** & **Speed of 90\%** & **Speed of 100\%** \\ \hline \multirow{4}{*}{4} & Second- & Aerodynamic & 4969.7 & 4968.6 & 4970.5 & 4966.2 \\  & order & Load & 5000.5 & 5008.6 & 5017.6 & 5028.4 \\  & torsion & Centrifugal and & & & & \\  & & Aerodynamic & 4995.2 & 5001.8 & 5012.7 & 5019.4 \\  & & load & & & & \\  & & Aerodynamic & 6195.6 & 6194.2 & 6194.3 & 6191.7 \\  & & bending & Centrifugal load & 6236.6 & 6248.6 & 6262.2 & 6277.6 \\  & & Centrifugal and & & & & \\  & & Aerodynamic & 6235.7 & 6246.8 & 6261.2 & 6275.1 \\  & & load & & & & \\  & & Aerodynamic & 6893.8 & 6894.0 & 6897.2 & 6894.6 \\  & & load & & & & \\  & torsion & Centrifugal load & 6999.0 & 7031.3 & 7068.0 & 7109.8 \\  & & Centrifugal and & & & & \\  & & Aerodynamic & 6996.4 & 7028.0 & 7067.4 & 7105.9 \\  & & load & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._administration, H.B., F.Z. and J.W. ; funding acquisition, H.B., F.Z. and J.W. All authors have read and agreed to the published version of the manuscript.