# Estimating Natural Frequencies of Cartesian 3D Printer Based on Kinematic Scheme
## 2 Vibration Analysis of 3D Printers
### Flexible Joint Model
The length of all vertical beams is equal to \(L_{1}\). The moments of inertia of the vertical and horizontal beams are equal, respectively, to:  
\[\theta(t)=\theta(0)e^{j\omega t}. \tag{7}\]6  
The kinetic energy of motion of the parallelepiped structure is equal to:  
\[\theta(t)=\theta(0)e^{j\omega t}. \tag{7}\]7  
potential energy:  
\[\theta(t)=\theta(0)e^{j\omega t}. \tag{7}\]8  
The Lagrange equation after transformations will be:  
\[\theta(t)=\theta(0)e^{j\omega t}. \tag{7}\]9  
The frequency of the parallelepipedal structure, in this case, is equal to:  
\[-J\omega^{2}+k=0,\]0  
_Taking into account several degrees of freedom._ Usually, 3D printers have two, three or more degrees of freedom, including oscillations in directions of \(X\) and \(Y\) axes, oscillations of the printbed independently from the frame, moreover, some printer designs provide horizontal and vertical elements of the frame to oscillate independently as well, like the open kinematic chain design shown in Figure A3b. If we use linearized models and also assume that the stiffness matrices in the Equation (3) are diagonal, treatment of multiple degrees of freedom is fairly simple: each rotational degree of freedom is considered as if  
\[-J\omega^{2}+k=0,\]1 Model of the parallelepiped. the other degrees of freedom are fixed. For example, let us calculate the gantry structure vibration in another plane, orthogonal to the plane considered before. The kinetic energy of motion of the gantry structure in this case is exactly the same  
\[-J\omega^{2}+k=0,\]2  
but the potential energy, due to the fact that upper joints are not involved in the oscillations, is  
\[-J\omega^{2}+k=0,\]3  
This results in the frequency:  
\[-J\omega^{2}+k=0,\]4  
To simplify this approach, we present an Algorithm 1 for calculating Cartesian structures. For example, for the gantry and parallelepiped structures considered before longitudinal beams are vertical, transverse beams are horizontal, so Formulas (10)-(12) could be easily derived without Lagrangians. ``` Input: \(\{m_{i}\}\), \(\{L_{i}\}\), \(\{k_{i}\}\). Output: \(\omega\). 1. Select the plane in which the vibration of the structure will be investigated. Select an appropriate degree of freedom. 2. Decompose the construction into primitives: 1. \(n\) longitudinal beams of length \(L_{i}\) with mass \(m_{Li}\), length \(L_{Li}\) and moment of inertia: \(J_{Li}=\frac{m_{Li}{L_{Li}}^{2}}{3}\). 2. \(m\) transverse beams and lumped masses with the distance from the base to the center of mass \(L_{Ti}\), mass \(m_{Ti}\) and moment of inertia \(J_{Ti}=m_{Ti}{L_{Ti}}^{2}\). 3. Calculate the stiffeners \(k_{j},j=1\ldots N\) (\(N\) is the number of joints). 4. Estimate the frequency using the formula: \[-J\omega^{2}+k=0,\]5

**Algorithm 1**The algorithm for calculating Cartesian structure natural frequency. ## 3 Analysis of Sample Designs

The current section describes the application of the proposed method for estimating natural frequencies of 3D printers to some practically valuable examples. ### Experimental Analysis of a Gantry 3D Printer

It is possible to find the first lower mode for an existing 3D printer using the proposed approach. A gantry design Anycubic i3 Mega was chosen as an example. A distinctive feature of this 3D printer is the presence of a relatively heavy printhead mounted on two horizontal cylindrical guides working as a whole due to the rigid coupling of their ends (see Figure 8). The length of the vertical beams is labeled \(L_{1}\), and the height at which the printhead is located is labeled \(L_{2}\). This design has three types of joints with different stiffness. The kinetic energy of motion of the gantry 3D printer is equal to:

\[-J\omega^{2}+k=0,\]6

potential energy of motion:

\[-J\omega^{2}+k=0,\]7

The Lagrange equation after transformations will look like this:

\[-J\omega^{2}+k=0,\]8

The natural frequency of oscillation of the 3D gantry printer can be expressed using the formula:

\[-J\omega^{2}+k=0,\]9

This formula can be also obtained using the Algorithm 1.