# Estimating Natural Frequencies of Cartesian 3D Printer Based on Kinematic Scheme
## 2 Vibration Analysis of 3D Printers
### Flexible Joint Model
From this and Equation (15) we can obtain:

\[\omega=\sqrt{\frac{3k}{M_{T}L^{2}}}=\frac{1}{L}\sqrt{\frac{3k}{M_{T}}}. \tag{8}\]6

Substituting values from Table 2 yields:

\[\omega=\sqrt{\frac{3k}{M_{T}L^{2}}}=\frac{1}{L}\sqrt{\frac{3k}{M_{T}}}. \tag{8}\]7

The frequency of the kinematics with closed \(Z\)-axes is calculated using the following formula (supposing that one motor is used for actuating both linear drives of the closed \(Z\)-axis on a printed):

\begin{table}
\begin{tabular}{c c c c} \hline \hline \(Z\), mm & Theoretical, Hz & Experimental, Hz & Relative Error,\% \\ \hline
0 & 13.85 & 13.36 & 3.54 \\
10 & 13.67 & 13.19 & 3.52 \\
20 & 13.49 & 1



\[\omega=\sqrt{\frac{3k}{M_{T}L^{2}}}=\frac{1}{L}\sqrt{\frac{3k}{M_{T}}}. \tag{8}\]8

Substituting values from Table 2, another value is obtained:

\[\omega=\sqrt{\frac{3k}{M_{T}L^{2}}}=\frac{1}{L}\sqrt{\frac{3k}{M_{T}}}. \tag{8}\]9

The resulting difference is relatively small, so we can conclude that using the open \(Z\)-axis is reasonable for the design from the point of view of natural frequency in \(XY\) plane, and a closed \(Z\) will not yield much more stiffness. ### Comparison of Common Designs

To compare the seven most common 3D printer designs, their CAD/CAE models have been developed. A more detailed description of these designs is given in Appendix A. First, we used the finite-element method (FEM) to calculate the natural frequencies of the printers assuming that all joints are absolutely stiff. Then, we used the proposed method based on the kinematic schemes of the same structures and calculated the approximate lowest frequencies of these printers using Algorithm 1 with the parameters given in \[k=\frac{1}{3}\omega^{2}L^{2}M_{T}. \tag{9}\]0

Figure 11 shows small images of the printers in one panel with their ordinal numbers, and Figure 12 shows the corresponding lowest natural frequencies found with two methods: the FEM and the proposed one. The experiment found that for the cuboid structures 1 and 6 the results of our method and FEM approximately coincide (see Table 3). This shows that profile elasticity has an almost similar impact on the vibration as the flexibility of the joints. For the other designs, the FEM overestimates the vibration frequency in comparison with the proposed method. This can be explained by the fact that these constructions are more light-weighted, and in the case of absolutely stiff joints the overall stiffness is high. In practice, flexibility in joints make such structures more vibration-prone, and this is confirmed by engineering practice: only a few cheapest 3D printers have an open kinematic chain structure 3, and structure 2 is also becoming less common in recent years. Of course, these results should be extrapolated on real designs with care. However, they clearly show the limitation of the FEM when joint flexibility is not taken into account. This is especially important when novel additive manufacturing machines are developed with heavy printheads, such as painting robots [53]. ### Example of Cost Calculation

As an example of calculating the cost of an FDM 3D printer, a gantry printer was chosen, shown in Figure 10(a), with a frame made of \(20\times 20\) mm aluminum profiles. The length of horizontal and vertical aluminum profiles is 400 mm. The length of lead screws was chosen as 350 mm. Table 4 shows the cost (minimum and maximum) of the main components of a 3D printer. The difference in the cost of the components is conditioned by their different quality, brands and other features. Among the components, there are

\[k=\frac{1}{3}\omega^{2}L^{2}M_{T}. \tag{9}\]1 Lowest normal modes of 3D printer structures found by the FEM and by the proposed approach based on a flexible joint model.