# Estimating Natural Frequencies of Cartesian 3D Printer Based on Kinematic Scheme
## 2 Vibration Analysis of 3D Printers
### Flexible Joint Model
An example of a 3D printer with closed \(X_{H}\), \(Y_{H}\), \(Z_{B}\) axes (see Figure 10(b)) is the Total \(Z\) Anyform 3D printer [64].These 3D printers differ only in the closed \(Z\) axis of the table, which affects the stiffness of the table. As a rule, 3D printers with a closed \(Z_{B}\) axis are often significantly more expensive, which is due to using more precise ball screw drives instead of a trapezoidal screw and is not fully determined by the printer kinematics. Table 14 shows two variants used in practice. The first variant is the one with closed axes \(X_{H}\), \(Y_{B}\) and open \(Z_{H}\). This variant is trendy in the 3D-printers market due to its cheapness and simplicity of construction. Examples of a th

Wanhao Duplicator i3 [65] and Anycubic i3 Mega [51] (see Figure A3a). The second variant with open \(X_{H}\), \(Z_{H}\) axes and open \(Y_{B}\) (see Figure A3b) is a simplified version of the previous model, and one of the cheapest variations of Cartesian 3D printers. An example of such a 3D printer is the Wanhao Duplicator i3 Mini 3D printer [66]. Examining the designs from Table A5, we found only one variant of the existing 3D printer with two actuated printed axes. It is Felix 3.0 [67], a 3D printer with open \(Y_{B}\), \(Z_{B}\) and closed \(X_{H}\) (see Figure A4). The manufacturer claims that the 3D printer can develop a fairly high printing speed thanks to this design compared to its counterparts. Three-dimensional printers with the designs shown in Tables A6 and A7 were not found. Probably, such designs are not quite suitable for 3D printing. Moving the printing surface along all three axes can affect print quality since the printbed with the printed detail is much heavier than the printhead. However, it is possible that designs of this type are or will be used for other additive technologies, where complete fixation of the printhead is required due to its size and mass. A recent study published the design of a prototype 3D printer with a fully fixed extruder. However, this printer is not Cartesian and is not commercially available [68]. Now, make a complete classification scheme of the existing Cartesian 3D printers. It is given in Table 14. The classification scheme shows each case's fixed and actuated axes and the closed and open axes for each actuated axis. Only seven of the 40 possible kinematics designs (8 interchangeable) were identified among existing designs. Table 14 shows that the actuated axes are often closed for greater structural stiffness. Most of the actuated axes relate to the printhead. Figure A7. Fixed axes \(X_{B}\), \(Y_{H}\), \(Z_{B}\) and actuated axes \(X_{H}\), \(Y_{B}\), \(Z_{H}\). Existing designs are 1 and 3. Figure A9: Fixed axes \(X_{H}\), \(Y_{H}\), \(Z_{B}\) and actuated axes \(X_{B}\), \(Y_{B}\), \(Z_{H}\). ## References

* [1] [PERSON]; [PERSON]; [PERSON] Experimental optimization of fused deposition modelling processing parameters: A design-for-manufacturing approach. _Procedia Manuf._**2017**, _10_, 791-803. * [2] [PERSON]; [PERSON]; [PERSON]; [PERSON] Optimization of FDM 3D printing process parameters using Taguchi technique. _IOP Conf. Ser. Mater. Sci. Eng._**2021**, _1168_, 012022. [CrossRef]
* [3] [PERSON]; [PERSON]; [PERSON]; [PERSON] A physical investigation of dimensional and mechanical characteristics of 3D printed nut and bolt for industrial applications. _Rapid Prototyp. J._**2022**. [CrossRef]
* [4] [PERSON]; [PERSON]; [PERSON]; [PERSON] Additive manufacturing a powerful tool for the aerospace industry. _Rapid Prototyp. J._**2021**, _28_, 87-100. [CrossRef]* [PERSON] (2014) [PERSON] General classification of robots. Size criteria. In Proceedings of the 2014 23 rd International Conference on Robotics in Alpe-Adria-Danube Region (RAAD), Smolenice, Slovakia, 3-5 September 2014; pp. 1-6. * [PERSON] (2014) [PERSON] _Implementation of Robot Systems: An Introduction to Robotics, Automation, and Successful Systems Integration in Manufacturing_; Butterworth-Heinemann: Oxford, UK, 2014. * [PERSON] et al.