# 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 \(\mathbf{\mathsf{N}}\) & **Mode 1, Hz** & **Mode 2, Hz** & **Mode 3, Hz** & **Proposed Method, Hz** \\ \hline
1 & 31.12 & 31.38 & 53.99 & 31.59 \\
2 & 42.97 & 45.50 & 95.37 & 24.14 \\
3 & 58.04 & 67.02 & 103.5 & 17.81 \\
4 & 60.62 & 84.89 & 117.4 & 24.83 \\
5 & 32.83 & 40.81 & 77.71 & 25.67 \\
6 & 31.45 & 36.12 & 60.58 & 36.72 \\
7 & 48.73 & 50.38 & 96.97 & 27.93 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Modes 1–3 of the printers obtained by the FEM and the lowest mode by the proposed method. obligatory ones used in any configuration of a 3D printer and not depending on the kinematic scheme: the power supply unit (PSU), the printbed, the printbed and the controller board. The economic effect of creating a 3D printer with the specific kinematic scheme increases when cheap obligatory components are used, because in this case a rail guide system, motors, etc. will affect the total cost of the printer much more significantly. In the case of using top-level obligatory components, a more stiff kinematic scheme will not result in relatively much higher expenses, so it should be preferred. The exception here is the rail guide system, which is a rather costly component, but it can be replaced with cheaper cylindrical guides in many cases. ## 4 Conclusions

The paper proposes a simple method for calculating 3D printer natural frequencies based on the kinematic scheme. It is applicable in the case of high stiffness of elongated elements such as profiles, guides, and beams. This method allows replacing elaborate finite-element analysis of the detailed model on a preliminary design stage. The accuracy of this approach is relatively high and feasible for engineering purposes; the provided examples confirm its practical applicability. Several most common designs have been compared via their lowest natural frequency, and the variant with the greater value of vibration frequency is preferred using this technique. Additionally, we give an example of calculating the FDM printer cost. We conclude that using expensive obligatory parts such as the power unit, the printhead and the controller, may reduce the relative cost of mechanical components and therefore make using a more stiff kinematic scheme more reasonable. An example of decision-making concerning the type of 3D printer is given. We also propose a classification scheme for the Cartesian mechanical systems used for additive manufacturing which considers variants of already existing structures and examples not yet applied to 3D printing. It can be used by developers of various additive machines, such as desktop 3D printers, architectural 3D printers, and others, to simplify the engineering process in a preliminary design stage. The algorithm for natural frequency estimation can be used for computer search of proper designs using optimization. A table of stiffness of various joint components obtained from the experiment can be developed which would help engineers to find optimal design solutions. To summarize, the paper makes the following contributions. First, the paper provides theoretical and practical confirmation of the applicability of the flexible joint model and the necessity to take bolted joint flexibility into account during finite element analysis, which is a non-trivial task in most moderns software packages such as Fusion 360. Second, the paper gives a simplified and handy algorithm for natural frequency estimation of the 3D printer which does not require the application of the Euler-Lagrange formula directly and is suitable for both manual and machine usage. Third, a classification scheme for Cartesian 3D printers is proposed using two criteria: the type of kinematic chain associated with each axis (open/closed) and the mobility of each axis' elements (actuated/fixed) connected to the print head and the print bed.