# Estimating Natural Frequencies of Cartesian 3D Printer Based on Kinematic Scheme
###### Abstract  
Nowadays, 3D printers based on Cartesian kinematics are becoming extremely popular due to their reliability and inexpensiveness. In the early stages of the 3D printer design, once it is chosen to use the Cartesian kinematics, it is always necessary to select relative positions of axes and linear drives (prismatic joints), which would be optimal for the particular specification. Within the class of Cartesian mechanics, many designs are possible. Using the Euler-Lagrange formalism, this paper introduces a method for estimating the natural frequencies of Cartesian 3D printers based on the kinematic scheme. Comparison with the finite element method and experimental validation of the proposed method are given. The method can help to develop preliminary designs of Cartesian 3D printers and is especially useful for emerging 3D-printing technologies. 3D printer; Cartesian kinematics; vibration analysis; additive manufacturing; mechanical design +
Footnote †: journal: Applied  
0000-0001-5000-0000-00001-5000]E. [PERSON]  
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concentrated in a small number of clusters and the level of international collaboration in them is weak [15]. There is a similar situation with publications on the theory of 3D printer design. This paper tries to introduce one more important theoretical aspect of additive manufacturing into scientific discourse. The printing rate of the 3D printer is one of the most critical consumer characteristics, directly affecting its user experience and economic returns. A recent study by [PERSON] et al. proposed that three main factors limit the building rate in FDM/FFF technology: the rate of filament feed, the rate of filament heating, and velocity of the printhead positioning [16]. The first and second factors have been addressed in several recent studies. For example, a novel rotatable extruder with a slot-shaped nozzle was designed to increase the filament feed rate [17], and the heating rate was sufficiently increased by introducing a laser heating system [18]. In the article [19], the authors propose to use fuzzy adaptive control for balanced thermal distribution during extrusion, in order to achieve the highest quality rapid prototyping. However, increasing the speed of printhead positioning is probably the most challenging problem in mass-market printers. Despite the technical opportunity of using rigid and heavy CNC machine frames providing better printing quality at high speeds, the design of 3D printers always implies a trade-off between mechanical stiffness and the printer cost, which should be as minimal as possible for economic reasons. This is especially crucial in FDM/FFF 3D printers since this technology is the most inexpensive and the most demanded in the 3D printer mass market. Therefore, many various designs are present on the market. Unfortunately, almost each of them suffers from certain flaws limiting the building rate sufficiently below the theoretical maximum determined by the properties of the printing material. The vibration characteristics of 3D printers and components directly affect the accuracy of the work, as reported in several studies, e.g., [20; 21; 22]. The main problem in FDM/FFF 3D printing originating from mechanical imperfectness is ghosting (ringing): the emergence of a repetitive pattern on the printed detail surface caused by mechanical vibration of the printer frame. This matter limits the maximal acceleration and speed available to a particular 3D printer [23; 24]. Another defect also caused by mechanical imperfectness is corner swell, an outgrowth of the deposed material on a corner [24; 25]. An example of these defects is given in Figure 1. They are negligible within a specific range of positioning settings, but printing errors become significant and practically unacceptable when they exceed particular values. Figure 1: Example of corner swell (upper box) and ghosting (lower box). Arrows mark typical variants of these defects.