# Estimating Natural Frequencies of Cartesian 3D Printer Based on Kinematic Scheme
## 2 Vibration Analysis of 3D Printers
### Flexible Joint Model
Using the Formula (13), we can find the relationship between the printhead position on the \(Z\)-axis and the natural frequency of oscillations. A testbench for vibration data acquisition was assembled to measure the vibrations of the head, bed, and frame of the 3D printer. The testbench includes the following components: NI PXI 1042 controller with the PXI-4461 module for collecting vibroacoustic signals, piezoelectric accelerometer IMV VP-4200, accelerometer power amplifier, and custom mount for accelerometer printed on a 3D printer, which allows placing the sensor parallel to any of three axes \(X\), \(Y\), \(Z\) (see Figure 9). A series of experimental data were recorded with a sampling rate of 1000 Hz. The program for recording data into a file was implemented in the LabVIEW environment. The data received from the accelerometer was processed in the MATLAB environment using the built-in FFT function. We investigated the dependence of the natural frequency of oscillation on the height of the printhead. Since the printhead and related components involved in \(Z\)-axis movement (guides, belts, mounts) have a total mass of 1.7 kg, which is greater than the mass of the vertical beams (0.7 kg), the location of the printhead on the \(Z\)-axis affects the 3D printer natural frequency, much. \[\omega=\sqrt{\frac{3k}{M_{T}L^{2}}}=\frac{1}{L}\sqrt{\frac{3k}{M_{T}}}. \tag{8}\]0 Anycubic i3 Mega kinematic scheme. In the experiment, we used the g-code to set a short-term extruder movement at each of the positions along the \(Z\) axis with a step of 10 mm, and the maximum height at which the printhead was positioned was 200 mm. Printer movement gave an impulse to the printer frame impact with the spectrum as close to that experienced by a 3D printer during printing as possible. The accelerometer was mounted on the \(X\) axis of the 3D printer in the right corner, see \[\omega=\sqrt{\frac{3k}{M_{T}L^{2}}}=\frac{1}{L}\sqrt{\frac{3k}{M_{T}}}. \tag{8}\]1 This position allows determining the frequency of frame vibrations affecting the printhead directly during operation. The total stiffness of the 3D printer at zero position was found using Equation (13) to validate the theoretical approach based on the flexible joint model. Further, the theoretical natural frequencies of vibrations were found. Figure 10 shows the dependence of the experimental and theoretical natural frequencies on the position of the printhead along the \(Z\) axis. Figure 10 shows that the proposed approach successfully predicts the first mode of the Anycubic i3 Mega. The relative error of the theoretical frequency was 2-6 percent (see Table 1). \[\omega=\sqrt{\frac{3k}{M_{T}L^{2}}}=\frac{1}{L}\sqrt{\frac{3k}{M_{T}}}. \tag{8}\]2 The first mode of the Anycubic i3 Mega 3D printer. \[\omega=\sqrt{\frac{3k}{M_{T}L^{2}}}=\frac{1}{L}\sqrt{\frac{3k}{M_{T}}}. \tag{8}\]3 (**a**) Model of the 3D printer Anycubic i3 Mega. A circle marks a place of the accelerometer mounting, an arrow marks the direction of axis sensitivity (**b**) Example of the accelerometer mounting. ### Preliminary Analysis of a New Design

This subsection shows how natural frequency can be calculated from the classification scheme for an arbitrary 3D printer. Suppose, natural frequencies of cubic printers depicted in Figure 10(a),b are predicted. From Algorithm 1 it follows that the natural frequency equals to

\[\omega=\sqrt{\frac{3k}{M_{T}L^{2}}}=\frac{1}{L}\sqrt{\frac{3k}{M_{T}}}. \tag{8}\]4

where \(k_{i}\) is \(i\)-th joint stiffness, and \(J_{i}\) is \(i\)-th moment of inertia. The formula for the frequency of open \(Z\)-axis design is:

\[\omega=\sqrt{\frac{3k}{M_{T}L^{2}}}=\frac{1}{L}\sqrt{\frac{3k}{M_{T}}}. \tag{8}\]5

where \(M_{\rm{G}}\) is the guide mass and \(M_{P}\) is the profile mass. Let the mass of a horizontal beam equal \(M_{h}=\rho_{P}L_{h}\), where \(L_{h}\) is the length of the horizontal beam, and \(M_{v}=\rho_{P}L_{1}\) is the mass of a vertical beam.