# Estimating Natural Frequencies of Cartesian 3D Printer Based on Kinematic Scheme
## 2 Vibration Analysis of 3D Printers
### Flexible Joint Model
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Component** & **Amount** & **Cost of 1 Piece, \$** & **Total Cost, \$** & **Cost Ratio** \\ \hline Aluminum profile & 2400 mm & 3–5 (1000 mm) & 7.2–12 & 6–3\% \\ Lead screw & 5 pc. & 2–5 & 10–25 & 8–6\% \\ Rail guide system & 4 pc. & 7–25 & 28–100 & 22\% \\ Stepper motor & 4 pc. & 5–15 & 20–60 & 15–13\% \\ PSU & 1 pc. & 5–20 & 5–20 & 4\% \\ Print bed & 1 pc. & 15–50 & 15–50 & 12–11\% \\ Print Head & 1 pc. & 15–80 & 15–80 & 12–18\% \\ Controller Board & 1 pc. & 30–100 & 30–100 & 23–22\% \\ \hline \hline All components excluding additional spare parts & & 130.2–447 & \\ \hline \end{tabular}
\end{table} 
Table 4: The cost of a 3D printer in Figure A3a. Further work will be dedicated to testing some non-existing designs and considering other decision-making criteria, such as the system tolerance to the inaccuracy of the 3D printer components, in addition to the natural frequency criterion. Conceptualization, A.K. and D.B. ; data curation, D.B. ; formal analysis, A.K. and L.S. ; investigation, E.K. ; methodology, A.K. ; project administration, A.K. and D.B. ; resources, A.K. and L.S. ; software, E.K. and A.K. ; supervision, A.K. and D.B. ; validation, L.S. and D.B. ; visualization, E.K. ; writing--original draft, E.K. and A.K. ; writing--review and editing, L.S. and D.B. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The authors are thankful to anonymous Referees for their insightful reviews and useful recommendations. The authors declare no conflict of interest. The following abbreviations are used in this manuscript:

* Computer Numerical Control
* PCI extension for Instrumentation

## Appendix A Classification of Cartesian Designs: Principles and Examples

The usefulness of a specific classification scheme for practice depends on whether it is accurate enough to systematize all the existing knowledge and is capable of accommodating new subjects [54]. Basics of kinematic scheme topology classification and methods for mobility investigation are reported in papers [55; 56; 57]. For 3D printing, a detailed classification of possible designs may be helpful for engineers at the early design stage to choose the most feasible kinematics, for sellers to organize their catalogs, for purchasers to identify their requirements more efficiently, for researchers to place their novel designs among existing ones. Cartesian 3D printers consist of two independent parts: the frame with the printhead mounted on it and the printbed.To distinguish between the coordinate axes of the printhead and the printbed, we denote them as \(X_{H}\), \(Y_{H}\), \(Z_{H}\) for the printhead and \(X_{B}\), \(Y_{B}\), \(Z_{B}\) for the printbed. These parts are separated in space by the printed part and move relative to each other. The system requires the mobility of 3D printer parts along three orthogonal axes to be functional. For example, if the coordinate system as a whole is stationarily attached to the printbed, then the printhead must move along the \(X_{H}\), \(Y_{H}\) and \(Z_{H}\) axes. Hereafter, for brevity, we will use the term \"axis\" not only for the direction in space but also to denote a set of mechanical elements (beams, guides) designed to move the printer's working body along a given coordinate axis. Kinematics of modern Cartesian printers implies variation of mobility and immobility of the axes. In our analysis, the actuated axis is designated (A) and the fixed axis is designated (F). Table 14 shows all possible combinations of the printhead and the printbed axis mobility. Each row in the table represents a kinematics variant with three actuated axes. Because creating each layer of the printed part takes place in the \(XY\) plane, the \(Ox\) and \(Oy\) axes of the printhead and the printbed are interchangeable. Thus, the third row is synonymous with the fourth row, the fifth row is interchangeable with the seventh row, and only one version of each pair will be considered further.