## 5 Conclusions  
In this study, the analytical method of the beam mode with variable cross-section is derived, and modal analysis of the crossbeam of a gantry machine tool based on the analytical method is carried out and the following results are obtained:  
1. Based on the Euler beam theory and transmission matrix method, the analytical method of the beam mode with variable cross-section is deduced. By dividing the crossbeam of the gantry machine tool into 27 segments according to the different cross-sectional shapes, the analytical solutions of the crossbeam mode are obtained, which are validated by the finite element method. Comparing with the FEM, the analytical method has advantages in terms of computational efficiency.
2. By using the analytical method due to its high efficiency, the influence of the structural parameters on the crossbeam mode is investigated. Increasing the linear density in  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Structural Parameter** & **Sensitivity of First-Order Frequency** & **Sensitivity of Modal Node Position** \\ \hline \hline Bending stiffness in the middle part of the crossbeam (BM) & \(0.5014\) & \(-0.0103\) \\ Linear density in the middle part of the crossbeam (LM) & \(-0.2378\) & \(0.2744\) \\ Bending stiffness in the end part of the crossbeam (BE) & \(0.0137\) & \(0.0112\) \\ Linear density in the end part of the crossbeam (LE) & \(-0.3049\) & \(-0.2669\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Sensitivities value.  
Figure 9: Sensitivities of the crossbeam mode to structural parameters.  
the middle part leads to the decrease in the modal node span of the crossbeam, while increasing the linear density in the end part leads to the opposite result. The bending stiffness in the end and middle of the crossbeam has a very low influence on the mode shape.
3. The sensitivities of various structural parameters to the crossbeam mode are obtained separately. The results show that the main factor affecting the first-order frequency of the crossbeam is the bending stiffness in the middle part of the crossbeam, followed by the linear density in the middle and end part. The bending stiffness in the end part of the crossbeam barely makes a difference to the crossbeam mode.
4. This study provides an efficient method for the crossbeam modal analysis, which is expected to be widely used in the design of the machine tool crossbeams. The next step is to optimize the structural parameters of the crossbeam. However, the analytical method of the crossbeam mode is only suitable for the analysis of low-order modes because Euler beam theory ignores the influence of shear deformation and the moment of inertia. Therefore, increasing the precision of the high-order mode of the crossbeam will be the future research focus.  
This study was initiated and designed by [PERSON] and [PERSON]; the derivation of formulas, numerical simulations, and writing the paper were completed by [PERSON] and [PERSON]; [PERSON] provided suggestions for paper writing and simulations. All authors have read and agreed to the published version of the manuscript. This study was partly supported by the National Natural Science Foundation of China (52275092).  
Not applicable.  
The data are part of an ongoing project.  
The authors declared no potential conflict of interest with respect to the research, authorship, and/or publication of this article. The authors declare no conflict of interest.