# Estimating Natural Frequencies of Cartesian 3D Printer Based on Kinematic Scheme
## 2 Vibration Analysis of 3D Printers
### How Cartesian 3D Printer Kinematic Scheme Affects Its Dynamics
The kinematic scheme determines the number of degrees of freedom and the function \(J(\mathbf{x})\). Let us show more exactly how dynamical error is related to mechanical movement. Consider a case where stiffness matrices are diagonal, so each equation of the system (3) is  
Figure 3: Examples of kinematic schemes for Cartesian 3D printers. Rectangles denote prismatic joints, circles denote spherical joints, a black triangle denotes the printhead, and a black parallelepiped denotes the printed. Panels (**a–c**) refer to functional kinematic schemes, while the panels (**d–f**) additionally show frame and guide joints that cannot be considered absolutely stiff. independent. Then, one rotational degree of freedom of a linearized system is described as follows. Let the system be affected by the torque \(\mathbf{f}_{\theta}=Ae^{i\Omega t}\), where \(\Omega\) is the frequency of vibrations induced by the 3D printer parts' translational movements. In case of small angles the function \(L\) is linear, so we imply \(e=L\theta\), \(K_{\theta}=k\), \(C_{\theta}=0\). The Equation (3) then reads:  
\[ \begin{cases}J\ddot{\theta}+k\theta=Ae^{i\Omega t}\\ \mathbf{e}=L\theta.\end{cases}  \tag{4}\]  
Substituting a known solution \(\theta(t)=\theta_{1}e^{i\omega t}+\theta_{2}e^{i\Omega t}\) into the Equation (4), where \(\omega=\sqrt{k/\gamma}\) is a natural frequency of the printer vibration, we obtain:  
\[\theta_{2}=\frac{A}{J(\omega^{2}-\Omega^{2})}. \tag{5}\]  
The error amplitude maximal value is \(e^{*}=L(|\theta_{1}|+|\theta_{2}|)\). The low-frequency vibrations caused by massive 3D printer parts movements satisfy the condition \(\omega^{2}>\Omega^{2}\). The Equation (5) explains the profit from the natural frequency enhancement: increasing the natural frequency lowers down the amplitude of the dynamical error. The way how real Cartesian systems are implemented suggests using more complicated structures than the ones shown in Figure 3. A typical Cartesian system can be illustrated by the gantry-type kinematics often used in architectural 3D printers or CNC milling machines. Its functional kinematic scheme is given in Figure 4a following literature on CNC machine design [48; 49]. Meanwhile, a more detailed kinematic scheme given in Figure 4b shows some issues with practical implementations. In addition to actuated prismatic joints, each axis is supplied with passive linear translation mechanisms for several reasons: to decrease the load on actuated joints, to prevent skew and jamming in actuated joints, and to improve the overall stiffness. The latter is important in the context of the current paper. A study for the CNC machine [49] confirms the importance of passive joint stiffness. The engineering practice of 3D printer designing implies using multiple additional passive joints as well. Several types of passive linear guides are common including round rods with ball bearing carriages, profiled rails with ball bearing carriages and aluminum profiles with wheeled carriages. Their properties such as backlashes, precision and price are highly variable and should be considered individually for each manufacturer and each design, and their application is also highly variable. For example, in inexpensive gantry 3D printers a brintbed with round rod guides is used in (Prusa Research, Prague, Czech Republic) and Anycubic i3 Mega (Shenzhen Anycubic Technology Co., Ltd., Shenzhen, China) 3D printers [50; 51], while Sovol SVO3 (Sovol 3D, Shenzhen, China) has a wheeled  
Figure 4: Practical implementation of a Cartesian system. Panel **(a)** shows a functional kinematic scheme of a Cartesian gantry CNC machine and panel **(b)** gives a detailed kinematic scheme of its implementation with actuated prismatic joints (shown with white rectangles) and passive rail guides (shown with grey rectangles). Circles denote passive spherical joints with internal stiffeners, a black triangle denotes the printhead, and a black parallelepiped denotes the printbed  
carriage on an aluminum profile [52].