## 3 Crossbeam of Gantry Machine Tool: Analytical Solutions  
### Crossbeam Properties and Cross-Section Parameters  
As described above, the crossbeam is an important component in the gantry machine tool. The structural and anti-vibration properties of the crossbeam have a direct impact on the stiffness and machining precision of the gantry machine tool [11]. Therefore, modal analysis should be carried out in the design of the machine tool beam. In this section, we apply the analytical method, which is derived in Section 2, to the modal analysis of the crossbeam. A five-axis machine tool, as shown in Figure 2, is mainly made up of a column, crossbeam, saddle, ram, and milling head. As can be seen in the figure, the crossbeam connects the columns and saddle, which receives the excitation from the milling head in the X and Y directions. Therefore, in the design of the crossbeam, the bending mode of the crossbeam in both directions should be computed and analyzed.  
Next, we focus on the properties of the crossbeam in the gantry machine tool. The crossbeam model and its profile are shown in Figure 3. The crossbeam material is structural steel with a density of 7850 kg/m\({}^{3}\), an elastic modulus of \(2.1\times 10^{11}\) Pa, and a Poisson's ratio of 0.3. It should be pointed out that the crossbeam model has been simplified to eliminate some features that have little influence on the modal parameters, such as chamfering and small holes. As the profile shown in Figure 3b, the crossbeam is welded by multiple metal plates. The crossbeam profile is composed of multiple feature surfaces, which gives the opportunity to use the equation in Section 1 by dividing the crossbeam. According to the shape of its section, 27 segments of the beam with 4 different section shapes can be divided. The position of each cross-section is marked in Figure 3b. The crossbeam division is shown in Figure 3c. Each divided segment beam can be considered as a beam with equal cross-section, and its length and position are marked in the figure. The crossbeam has four different cross-sectional shapes, as shown in Figure 4. It should be noted that there are several segment beams with the same cross-sectional shapes. For example, the 1 st and 27 th segment beams share Section-a; the 2 nd and 6 th segment beams share Section-b; the 3 rd and 7 th segment beams share Section-c; the 4 th and 8 th segment beams share Section-d.  
Figure 2: Three-dimensional model of a five-axis machine tool.  
When calculating the moment of inertia of different cross-sections, it is necessary to take into account that the centroid position of each cross-section is different. The parallel-axis formula is used to modify the moment of inertia of different cross-sections to the axis where the center of mass of the crossbeam is located. The parallel-axis formula is as follows:  
\[\begin{array}{l}I_{x^{\prime}}=I_{x}+{a_{x}}^{2}A\\ I_{y^{\prime}}=I_{y}+{a_{y}}^{2}A\end{array} \tag{14}\]  
where \(I_{x}\) and \(I_{y}\) are the moments of inertia through the centroid, \(I_{x^{\prime}}\) and \(I_{y^{\prime}}\) are the moments of inertia through the \(X\) and \(Y\) axes, \(a_{x}\) and \(a_{y}\) are the distances from the centroid to the \(X\) and \(Y\) axes, and \(A\) is the cross-sectional area.