# Estimating Natural Frequencies of Cartesian 3D Printer Based on Kinematic Scheme
## 2 Vibration Analysis of 3D Printers
### Flexible Joint Model
Also, two mounting options for the actuated axes were considered:Open axes (\(A_{O}\))--actuated axes that have one attachment point to a fixed base or elements of other axes. This corresponds to the open kinematic chain. * Closed axes (\(A_{C}\))--actuated axes that have at least two attachment points to a fixed base or elements of other axes. This corresponds to a closed kinematic chain. The closed axis provides more rigidity to the construction but needs a more complicated design with additional guides, motors, etc. ### Classification of Cartesian Designs

In this subsection, we will overview all possible combinations of fixed and actuated axes and make a complete classification of Cartesian 3D printers. An example of the visual representation of each kinematics is presented in Appendix A in Figures A5-A10. It should be noted that the kinematic schemes can be depicted differently depending on where the entire structure is mounted (e.g., standing on the table or attached to the wall). All illustrated constructions shown in Appendix A are fixed on the floor. Tables A2-A7 show kinematic schemes with different variations. Since the \(X\) and \(Y\) axes are interchangeable, designs numbered 2 and 8 (marked yellow (*)) as well as 6 and 7 (marked red (**)) in some tables are the same, and one solution from each pair can be omitted. In Tables A4 and A5, there are no interchangeable designs because the axes \(X_{B}\) and \(Y_{H}\) (or \(X_{H}\) and \(Y_{B}\)) belong to different 3D printer components. \begin{tabular}{l c c c c c c} \hline \hline \(\mathbf{N}\) & \(\mathbf{X_{B}}\) & \(\mathbf{Y_{B}}\) & \(\mathbf{Z_{B}}\) & \(\mathbf{X_{H}}\) & \(\mathbf{Y_{H}}\) & \(\mathbf{Z_{H}}\) \\ \hline
1 & \(F\) & \(A_{O}\) & \(F\) & \(A_{O}\) & \(F\) & \(A_{O}\) \\
2 &. & \(A_{C}\) &. & \(A_{O}\) &. & \(A_{C}\) \\
3 &. & \(A_{C}\) &. & \(A_{C}\) &. & \(A_{O}\) \\
4 &. & \(A_{C}\) &. & \(A_{C}\) &. & \(A_{C}\) \\
5 &. & \(A_{O}\) &. & \(A_{O}\) &. & \(A_{C}\) \\
6 &. & \(A_{C}\) &. & \(A_{O}\) &. & \(A_{O}\) \\
7 &. & \(A_{O}\) &. & \(A_{C}\) &. & \(A_{O}\) \\
8 & \(F\) & \(A_{O}\) &. & \(A_{C}\) & \(F\) & \(A_{C}\) \\ \hline \hline \end{tabular} 

**Table A5**. Fixed axes \(X_{B}\), \(Y_{H}\), \(Z_{H}\) and actuated axes \(X_{H}\), \(Y_{B}\), \(Z_{B}\). \begin{tabular}{l c c c c c c} \hline \hline \(\mathbf{N}\) & \(\mathbf{X_{B}}\) & \(\mathbf{Y_{B}}\) & \(\mathbf{Z_{B}}\) & \(\mathbf{X_{H}}\) & \(\mathbf{Y_{H}}\) & \(\mathbf{Z_{H}}\) \\ \hline
1 & \(F\) & \(A_{O}\) & \(A_{O}\) & \(A_{O}\) & \(F\) & \(F\) & \(F\) \\
2 &. & \(A_{C}\) & \(A_{C}\) & \(A_{O}\) &. &. \\
3 &. & \(A_{C}\) & \(A_{C}\) & \(A_{C}\) &. &. \\
4 &. & \(A_{C}\) & \(A_{C}\) & \(A_{O}\) &. &. \\
5 &. & \(A_{O}\) & \(A_{C}\) & \(A_{O}\) &. &. \\
6 &. & \(A_{C}\) & \(A_{O}\) & \(A_{O}\) &. &. \\
7 &. & \(A_{O}\) & \(A_{O}\) & \(A_{C}\) &. &. \\
8 & \(F\) & \(A_{O}\) & \(A_{C}\) & \(A_{C}\) & \(F\) & \(F\) \\ \hline \hline \end{tabular} 

**Table A6**. Fixed axes \(X_{H}\), \(Y_{H}\), \(Z_{B}\) and actuated axes \(X_{B}\), \(Y_{B}\), \(Z_{H}\). \begin{tabular}{l c c c c c c} \hline \hline \(\mathbf{N}\) & \(\mathbf{X_{B}}\) & \(\mathbf{Y_{B}}\) & \(\mathbf{Z_{B}}\) & \(\mathbf{X_{H}}\) & \(\mathbf{Y_{H}}\) & \(\mathbf{Z_{H}}\) \\ \hline
1 & \(A_{O}\) & \(A_{O}\) & \(A_{O}\) & \(F\) & \(F\) & \(F\) & \(A_{O}\) \\
2 * &with closed axes \(X_{H}\), \(Y_{H}\), and \(Z_{H}\) are VORON 2.4 [62] and a custom printer with Core-XYZ kinamtics [63]. Creating 3D printers with bed fixation is sometimes not a convenient technical solution but rather a forced one. The choice of this configuration for architectural printers can be explained by the fact that we can not move the platform on which the building stands. Table 13 illustrates two variants used in practice. Both variants are the most popular among all designs. An example of a 3D printer with closed \(X_{H}\), \(Y_{H}\) and open \(Z_{B}\) axes (see Figure 10(a)) is Anycubic 4 max Pro [51] or Ultimaker 3.