## 4 Influence of Structural Parameters on Crossbeam Mode  
### Structural Parameters Analytical Solutions  
The influence of various structural parameters on crossbeam modes is discussed in this section. Prior to that, however, it should be noted that, in order to avoid repeated interpretation, only the Y-direction mode of the crossbeam is analyzed in this section due to the similarity of the influence of structural parameters on crossbeam modes in both directions. It can be obtained from Equation (1) that the crucial factors affecting the crossbeam mode are linear density and bending stiffness. In addition, the vibration of the gantry machine tool is transmitted from the saddle to the crossbeam and then to the column at the sides. Thus, in the crossbeam design, we pay more attention to the structural parameters in the end and middle parts of the crossbeam. The research in this section focuses on the influence of linear density and bending stiffness in the end and middle part of the crossbeam on the bending mode. As can be seen from Figure 3, cross-sections with different shapes are staggered in the crossbeam profile. In order to analyze the influence of structural parameters in the ends and middle part, the crossbeam model should be treated to some extent. Hence, the 27-segment crossbeam model is equivalent to the 3-segment crossbeam model, where the end part segments include segments 1 to 7 and 21 to 27, and the middle part segment includes segments 8 to 20. The linear density and bending stiffness in the end part and middle part are linearly averaged and the correction coefficient \(k^{\prime}\) is taken to modify the 3-segment crossbeam model. The correction expression is as follows:  
\[ \begin{array}{l}\left(\rho A\right)_{eq}=k^{\prime}_{1}(\frac{1}{\sum_{i}} \cdot\sum_{i}\left(\rho A\right)_{i}l_{i})\\ \left(EI\right)_{eq}=k^{\prime}_{2}(\frac{1}{\sum_{i}}\cdot\sum_{i}\left(EI \right)_{i}l_{i})\end{array}  \tag{15}\]  
The comparison of the results of the 3-segment crossbeam model and the 27-segment crossbeam model is shown in Table 2. The corrected model has a low error, so the 3-segment crossbeam model can replace the 27-segment crossbeam model in this analysis. After validating the analytical method, we next analyze the crossbeam mode to explore the influence of the various structural parameters on the mode. The trend of crossbeam mode with the structural parameters in the middle part of the crossbeam is shown in Figure 7. The arrow direction in the figure represents the increase in the structural parameter. It can be seen from Figure 7 that the first-order natural frequency of the crossbeam increases \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Direction** & **Parameter** & **Analytical** & **FEM** & **Error/\%** \\ \hline \multirow{3}{*}{X} & First-order bending & \multirow{3}{*}{144.88} & \multirow{3}{*}{146.77} & \multirow{3}{*}{\(-\)1.29\%} \\  & frequency/Hz & & & \\ & Left modal node & & & \\ & position/mm & & & \\ \hline \multirow{3}{*}{Y} & First-order bending & \multirow{3}{*}{189.82} & \multirow{3}{*}{181.99} & \multirow{3}{*}{4.3\%} \\  & frequency/Hz & & & \\ & Left modal node & & & \\ & position/mm & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Analytical mode and FEM mode. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **27-Segment Beam** & **3-Segment Beam** & **Error/\%** \\  & **Model** & **Model** & **Error/\%** \\ \hline First-order bending & \multirow{3}{*}{189.822} & \multirow{3}{*}{188.5133} & \multirow{3}{*}{\(-\)0.689\%} \\ frequency/Hz & & & & \\ Left modal node & & & \\ position/mm & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of 3-segment beam model and 27-segment beam model. obviously with the increase in the bending stiffness in the middle part. The linear density in the middle part of the crossbeam has the opposite trend.