The Effect of 3D Printer Head Extruder Design on Dynamics and Print Quality

Abstract

In 3D-printing variations using FFF technology, the extruder governs printer efficiency. One of the important parameters is its weight, which affects the dynamics of the print head. Heavy print heads lead to high inertial forces and vibrations, limiting the printing speed and accuracy. The presented extruder solution is based on calculations of all its key elements and a simulation of heat distribution. The extruder with reduced dimensions and weight compared to the competing solutions was presented. This slightly affects the dynamics of the extruder head, which can be improved by adding passive control elements to the extruder driving system. The extruder head, being 20% lighter, allowed for about a 2% decrease in its displacement under the applied load course. This design allows for a lighter 3D printer head overall, thereby reducing its inertia and ensuring proper performance. This study presents the effect of using a 3D printer with a lighter extruder on the quality of the bearing bushings printed for ball joints. The ball joint bushings printed on the unmodified 3D exhibited a 50% damage rate, while, in the case of the modified 3D printer, the damage rate was lowered to 14%.

1. Introduction

Currently, 3D printing, also known as additive manufacturing, is increasingly being used. However, according to [1], these two terms should not be used interchangeably, since 3D printing refers to small-scale, at-home printing operations. Additive manufacturing (AM) refers to large-scale or industrial manufacturing.

Vidler et al. [2] stated that AM is a growing interdisciplinary area that covers applications such as medical devices [3,4], aerospace parts [5,6], microfabrication techniques [7,8], and artificial organs [9,10,11]. Among AM methods, light-based printing technologies such as two-photon polymerization [12,13], projection micro-stereolithography [14,15,16], and volumetric printing [17,18,19] have attracted considerable interest due to their speed, resolution, or uses in bio-fabrication. Three-dimensional (3D) printing technology, which produces objects through layer-by-layer material deposition directly from a numerical model, has become increasingly popular in various industries [20,21].

Vidler et al. [2] presented a 3D-printing technique utilizing an acoustically modulated, constrained air–liquid interface to swiftly create centimeter-scale 3D shapes.

Since 2009, 3D printing has been increasingly used for prototype production, as well as single- and low-volume production [22].

According to [23], Fused Deposition Modelling (FDM) and FFF are the dominant methods used in 3D printing. They come in varied sizes and shapes, but always include the following:

• Kinematics (a system that controls the movement of mechanical parts);
• An extruder;
• A build plate (print bed);
• Various electrical and electronic components (e.g., mainboards and power supplies);
• Auxiliary components of the system (frame, and build plate adjusters).

The 3D-printing method differs depending on the device; 3D printer kinematics enable the control of extruder movements.

The 3D printers can be classified according to their movement mechanisms, falling into categories such as 3D Cartesian, Delta, Polar, and SCARA. In addition to 3D printers, some companies employ industrial-scale robotic manipulators. These devices are distinct from those described above and exhibit significant variations among themselves. The movement of a 3D printer’s print head along the X- and Y-axes, while the print bed moves vertically along the Z-axis, is characteristic of the most popular Cartesian-style printers.

The 3D printer extruder motion dynamics can be better controlled using models and control loops to precisely manage the material flow by dynamically adjusting the motor speed and pressure based on real-time measurements. This goes beyond simple pre-programmed movements to account for variations in the material, speed, and temperature, ensuring the accurate deposition by correcting issues such as over- or under-extrusion. Common techniques include force-controlled printing to maintain a constant filament pressure and advanced feedforward controllers, often using PID loops, to respond to transient errors [24,25,26,27].

According to [28], the quality of the 3D-printed models is greatly affected by the design of the device and software configurations. The mechanical characteristics of the device and the setting of the printing parameters can influence the precision and consistency of printing.

The quality of the 3D prints can fluctuate based on various factors unique to each print, including the filament type, printing speed, and nozzle diameter. Closed-loop process control techniques enhance the precision and consistency of 3D printing [29].

Singhal et al. [30] highlighted that a flawless 3D-printing process, specifically regarding material extrusion (MatEx)-based additive manufacturing (AM), requires a methodical approach that takes into account all process parameters, including the nozzle diameter, nozzle angle, and printing speed.

The management of 3D printer temperatures entails the accurate regulation of the nozzle and bed heat to enhance the print quality, layer bonding, and mechanical characteristics. For common materials such as PLA, this means nozzle temperatures of approximately 190–210 °C and bed temperatures between 50–60 °C. Advanced applications employ enclosed heated environments, additional heating tools (hot air and IR), and even phase-change materials (PCMs) to ensure consistent conditions for high-performance materials like ABS (acrylonitrile butadiene styrene) or PEEK (polyether ether ketone), or for sensitive applications in areas such as the aerospace or medical fields [31,32].

1.1. Materials Used in FDM

The materials used for FDM exhibit resistance to UV radiation, hardness, translucency, biocompatibility, and other notable characteristics. They are used in AM processes that require a continuous filament of a thermoplastic material as an input. The 3D printer analyzed in this study was predestined for materials belonging to the Thermoplastic Polymer class [33].

1.2. Classes of Cartesian FFF Machines

Sharma and Patterson [21] reported that the x–y–z frame system is by far the most widely used 3D-printing system. It is more affordable, easier to use and maintain, and significantly more user-friendly for most users than the previous version. Four distinct classes were developed for FFF machines with Cartesian frames. The coordinate systems used in the four FFF frame classes (based on the information from [34]) are presented in Figure 1. The arrows indicate the movement directions of head 1 and table 2.

1.3. Direct FFF 3D Printer Extruder

The direct extruder [20] is characterized by the compact location of all the elements included therein. The drive unit that pushes the filament is positioned as close as possible to the print head. The drive unit and head are combined into a single unit that moves during printing. The small distance between the knurl and head did not cause excessive resistance when the filament moved from the drive unit to the print head. Additionally, it allows for easier control over the extrusion process or material retraction during idle movements. The short length of the compressed filament fiber does not cause a buckling effect in the channel, allowing for the use of a wider range of flexible materials for printing, such as TPU. A characteristic feature of stepper motors used in the construction of 3D printers is their high weight. Such motors are also used to drive extruders, resulting in direct-type heads weighing up to 400 g, without cooling fans, blowers, and holders. The high weight limits the printing speed, facilitating the loss of steps by drive axis motors. The consequence of using high accelerations for axes with high head inertia is the generation of unfavorable vibrations, which worsen the print quality.

The design solutions of direct-type extruders available on the market differ from each other in technical features, such as weight, size, or maximum extrusion force. The initial implementations of direct-type extruders, such as the MK8 type [35], had a problem with heat dissipation, which was partly dissipated by the radiator, but also transferred to the aluminum housing of the drive motor. The knurl that pushes the filament is located on the motor shaft. This requires a high-torque motor, which increases the mass of the entire structure. Over time, solutions that separated the drive unit from the plasticizing head were developed [36]. Recently, extruders with single- or multi-stage reducers resembling a single block in appearance have appeared on the market. For example, the Hemera HS extruder [37] possesses a block that includes a plasticizing head with a cooling system, pressure roller, reducer, and drive motor. These solutions have refined cooling systems that prevent heat conduction from the radiator to the drive motor. The use of a gear reducer significantly increased the extrusion force and enhanced the motor resolution (i.e., the accuracy of material dosing). There are solutions on the market, such as the BIQU H2 extruder [38], which has a very good ratio of mass to extrusion force; at a weight of 211 g, it can generate an extrusion force of 73.6 N. Heavier solutions with higher extrusion forces are also available. To enable very high printing speeds (up to 1000 mm/s) at high acceleration values, small extruders were created without dedicated plasticizing heads using small pancake-type stepper motors in the Nema 14 standard. Their characteristic feature is their being extremely lightweight (without a plasticizing head) while maintaining a high extrusion force. The flagship example of such an application is the Sherpa Mini extruder [39], which weighs up to 110 g (without the head), and the force that can be exerted on the filament fiber is 55 N.

In the present study, a prototype of the direct extruder was considered in two versions, namely, before and after modification.

1.4. The 3D-Printing Process Problems That Affect the Quality of the Resulting Products

During the FF process, the extrusion force increased with increasing mass flow. At high printing speeds, the material barely reached the plasticizing temperature. The plasticized material must overcome both shear and geometric resistances, significantly increasing the force required to drive the extruder at higher printing speeds [40]. The high weight of a direct extruder drive system causes high inertial forces to load the entire printer head assembly, which limits both the maximum speed v and the maximum acceleration a in the direction of movement. This weight can be significantly reduced during the design phase by optimizing the relative strength (strength per density unit) of the head components, particularly the movable components. As the kinematic friction coefficient decreases with increasing normal force in contact with surfaces under dry or boundary friction conditions [41], high acceleration values can introduce undesirable vibrations that affect the quality of the final printout [35].

Printing speed is one of the most important features of 3D printers, as it directly affects their comfort of use and cost-effectiveness [42]. Three main factors limit the building speed in FFF technology: extrusion, plasticization, and head positioning speeds. The first two factors typically refer to the extruder parameters. The third is related to the frame and kinematic system. Increasing the speed of the printing head positioning is probably the biggest problem with printers. Despite the technical possibility of using rigid and heavy machine frames that provide a better print quality at high speeds, designing 3D printers is always associated with a compromise between mechanical stiffness and the cost of the printer, which, for economic reasons, should be minimized. This is especially important in FFF 3D printers because this technology is the cheapest and most desirable in the mass market of 3D printers [42].

Vibrations from 3D printers directly affect the accuracy of 3D printer operations [42]. The main problem resulting from the mechanical imperfections of the connections was the ghosting effect, which consisted of manufacturing errors on the printed surface. Most often, when the direction of the head movement changes as a result of its inertia, mechanical vibrations of the frame occur. This issue limits the maximum acceleration and printing speed of a specific 3D printer model. The size of the vibrations that occur directly depends on the stiffness of the structure, which is influenced by several other factors. Nevertheless, reducing the amplitude of system vibrations and increasing the natural frequency can improve the quality of 3D printing [42]. For the cuboid structures of 3D printer frames, the natural frequencies were found to be the highest, which makes the aforementioned structures much better suited for applications in precise printing with a high accuracy [42]. Therefore, in the present study, the prototype 3D printer possessed a cuboid structure in its frame.

In [42], the frame connections were assumed to be rigid, but, in practice, their flexibility increases their vibration susceptibility. Engineering practice confirms this, as only a few low-cost 3D printers still use open kinematic chain structures, which are now rarely applied.

Wang et al. [43] reported that low-end FDM machines often suffer from poor dimensional accuracy due to multiple error sources. They proposed a three-step error compensation strategy to characterize extruder positioning errors and the Huang et al. [44] method to model the shape deformation. The experimental results confirmed the effectiveness of this approach in improving the dimensional accuracy of FDM processes.

Emelyanov et al. [45] reported that the vibration’ level of the construction 3D printer trolley was within the standard values. The vibration of the equipment was affected by the elasticity coefficient of the mixture. With an increase in such a coefficient, the vibration intensity of the equipment increases.

As AM advances, the related shortcomings and drawbacks of the processes become increasingly apparent. Creating and refining new processes can be costly, which makes it much more efficient and effective to focus on solving problems with existing processes and improving them [21].

Stopka et al. [46] observed an increase in the precision of 3D printers and the printing of small parts. The authors proved that successfully building a 3D printer of large dimensions required a dynamical simulation of the moving parts of a 3D printer to find suitable types of powertrain motors.

According to Achrén and Bårdén [47], FDM print quality can suffer from design flaws, poor construction, or cheap components; however, the input-shaping method helps reduce these issues and eliminates the “ringing” defect. Tests on cantilever 3D printers showed that input shapers allowed higher speeds and accelerations, significantly improved the surface roughness, and removed artifacts, enabling faster and more reliable printing.

Similar to many mechatronic and coordinate-based manufacturing systems [34,48], the mechanical compliance and vibrations present in an FFF machine can create defects in the produced parts. This may harm or ruin them, or, at the very least, compromise their dimensional precision and stability [49,50]. Although errors are probably found, to some degree, in all types of FFF machines, these irregularities are most prevalent at elevated speeds and high-acceleration levels. The use of faster print speeds is crucial for significantly lowering the cost of AM components and broadening their applications; however, the potential for errors remains a concern. Overshooting errors are frequently encountered in FFF machines, and they are noticeable to the naked eye. The deviation is significant, as indicated by the observation that the deviation error is greater than the full thickness of the part shell in some areas [21].

Another major origin is the friction present within the system. Minor vibrations caused by friction within the system may pose a problem, although their impact would likely go unnoticed without a thorough inspection of the parts. Compared to the extent of mechanical vibration in the system, the vibration produced by friction is expected to be minimal [51]. Consequently, Sharma and Patterson [21] overlooked friction vibrations to concentrate on the errors induced in the frame and belts.

Interestingly, Rossi et al. [52] studied a potential method to improve the quality of 3D-printed materials by inducing vibrations in the extrusion nozzle of an FT-5 R2 3D printer. This was performed to increase the inter-bead and interlayer adhesion throughout the printed parts. It was found that the induced vibrations induced only a small deviation from those caused by the printer itself.

The 3D-printing industry is experiencing an increasing need for faster printing speeds and improved precision [53]. However, as the velocity increases, the error also increases. The most efficient method for minimizing errors in the final position is the use of proper control systems [21].

Implementing a higher printing speed requires the optimization of the weights of the movable components. Heavy parts require an unnecessary increase in the engaged power of the driving motors. Parts that are too light are more susceptible to vibration induction in the mechanisms. In such cases, it is necessary to ensure the isolation or damping of these vibrations by design or to compensate for them by appropriately adjusting the control algorithms of these motors, which can be costly. There is also a trend toward a general reduction in the weight of machines and devices [54,55]. Therefore, the new prototype 3D printer consisted of elements made of aluminum alloys.

It is also necessary to provide a proper temperature for the realized process, which requires the involvement of heating devices and sufficient cooling. The latter can be realized using a set of radiators or a fan. Using the latter in some arrangement can create Coriolis forces that negatively affect the motion of the extruder carriage.

1.5. Analyzed Prototype Printer

The analyzed prototype printer (Figure 2) falls into the class of Makerbot-frame printers (Figure 1b), which are usually used as medium-to-large commercial-grade machines. The movement of the print bed is limited along the Y-axis (building direction), whereas the extruder carriage encompasses the X- and Z- axes.

This design provides a substantial build area and facilitates the easy integration of several extrusion nozzles; however, it is significantly slower than Prusa-frame machines (500–1000% in certain instances). In the Prusa-frame configuration, the print bed is fixed to move only along the Z-axis, whereas the extruder carriage moves along the X- and Y- axes. It is usually responsive to the filament quality and requires additional maintenance between prints because of the “hanging” cantilever build plate [21].

The first configuration of the prototype was based on the technology of assembled components manufactured using various techniques, including machining, casting, and plastic-processing techniques. Therefore, it was necessary to use, among others, reinforcing ribs and many connecting screws.

The modified version of the prototype was more compact and unified in terms of material (an Al alloy was used), and unnecessary connecting screws were eliminated, resulting in a reduction in the weight of the entire extruder assembly. Therefore, among others, a multi-mass dynamic model of a prototype 3D printer was developed according to our concept.

The goal of this study was to evaluate the effect of the modifications applied to the extruder, especially the reduction in its mass and the use of a fan, on the dynamics of the extruder carrier compared with the first version of the extruder. A lighter head enabled easier (faster) position compensation relative to the given head trajectory.

2. Materials and Methods

The design and modification of the extruder were performed considering the results of a thorough analysis of the printer components. The presented solution of the extruder is based on calculations of all its key elements based on the printer’s dynamic model and carriage simulation model for thermal analysis.

2.1. Extruder Design

The direct-type extruder often comprises a single-stage gear reducer with straight teeth, providing a small extruder weight while maintaining a high value of extrusion force and higher accuracy of filament feed. The reliable operation of a straight-toothed gear requires that the distance between the axles of the mating wheels is within the appropriate tolerances. Hence, the resulting clearance between the teeth directly affects the precision of the material dosing during printing. Providing the appropriate and tolerated distance between the bearing bush holes becomes more difficult with smaller gear elements. In straight-toothed two-stage gear reducers, the clearance between the teeth usually does not exceed 0.1–0.2 mm. This clearance can be eliminated, as shown in Figure 3.

The swinging montage of the motor with the pinion on its shaft allows for the regulation of clearance for minimization. This is of great importance in the case of a retraction procedure, which is an indispensable part of the printing program. It consists of withdrawing the filament from the head by the opposite rotation of the extruder drive shaft, while the printer head moves in the working space of the table (hotbed), thus avoiding unwanted leakage of the plasticized filament from the nozzle. The use of retraction allows for a blemish-free printout in the form of thin-filament threads. Figure 4 presents the 3D model of the designed extruder that allows the development of an extrusion force of up to 74 N. The optimal thermal behavior of the extruder unit during the printing process can be defined as the temperature of the Cold-End (the radiator), which, in this case, is the body of the extruder, and must be close to the ambient temperature, while the temperature of the so-called Hot-End (the plasticizing head) must be stabilized at the plasticizing temperature. From an economic perspective, it would be best if the heat generated by the electric heater was only applied to the plasticized material. This heat is transferred to the environment through the contact of the heating block with air or through the throat to the heat sink. Therefore, the throat was made of stainless steel with a low thermal conductivity coefficient of 14 W/(m·K) to minimize heat loss. The temperature distribution in the radiator body can be determined by simulating the heat transfer between individual components in the extruder assembly.

2.2. Exemplary Effect of the Modification of the Extruder Head in the Prototype 3D Printer

The need to modify the head in the process of constructing a prototype printer was supported by the following observations made during the preliminary tests of the products obtained from such a printer. An example of a drop in the quality of printouts using the non-optimized 3D printer extruder head was observed when printing bushings intended for testing ball joints subjected to multiaxial loads on a tribo-tester (Figure 5). Because the original ball joint bushings were not available on the market, replacements with the same geometry and material as those closest to the original ones were necessary. Before the modification of the extruder head, using a prototype 3D printer based on polymer filament, a series of ball joint bushings were used for testing ball joints. The bushings were printed using ABS, although PET-G and PLA were also used. The geometry of these ball joint bushings matched that of the vehicle manufacturer’s bushings.

The printer extruder head used before modification did not provide an optimal temperature distribution. This was due to the relatively heavy head that promoted heat overaccumulation in the head material, and the extruder fan used was too weak to dissipate the required amount of heat.

During the assembly of the printed bushings in the body of the ball joint mounted in the measuring head of the tribotester, the printed bushings were deformed or broken in some cases. For bushings printed on a 3D printer with the extruder head before modification, damage occurred in 50% of the printouts.

Interestingly, after the modification of the extruder head, damage occurred less frequently. Examples of correctly made printouts are shown in Figure 6a, while damaged printouts are shown in Figure 6b.

2.3. Model of the Prototype 3D Printer Extruder Carriage System

Because print quality is influenced by the extruder mass, the use of an extruder head proved beneficial.

The mathematical model of the device was the basis for the simulation analysis of the entire facility. The complexity of the device structure implies that the implementation of a full simulation requires appropriate simplifying assumptions. The effect of temperature on the deformation of printer components, including their stiffness and damping coefficients, was not considered in this study. Residual stresses that could arise during the welding of the frame elements were also not considered. The dynamics of the motion of the prototype 3D printer extruder carriage system was analyzed using the multibody model presented in Figure 7, similar to the model used in the simulations carried out by Sharma and Patterson [21].

The equations of motion of the moving components of the system are described by the formulae presented below. The motion of the printer mount with mass $m_1\left[\mathrm{k}\mathrm{g}\right]$ was described by Formula (1):

$${\ddot{x}}_1=-{\displaystyle \frac{\left(k_1+k_2\right)}{m_1}}{x}_1+{\displaystyle \frac{k_2}{m_1}}{x}_2-{\displaystyle \frac{\left({\beta }_1+{\beta }_2\right)}{m_1}}{\dot{x}}_1+{\displaystyle \frac{{\beta }_2}{m_1}}{\dot{x}}_2$$

(1)

where:

• ${\ddot{x}}_1$—printer mount acceleration [m/s²];
  ${\dot{x}}_1$—printer mount velocity [m/s];
  $x_1$—printer mount displacement [m];
  $k_1$—stiffness between ground and printer mount [N/m];
  $k_2$—stiffness between printer mount and printer frame [N/m];
  ${\beta }_1$—coefficient of energy dissipation between ground and printer mount [N × s/m];
  ${\beta }_2$—coefficient of energy dissipation between the printer mount and printer frame [N × s/m].

The motion of the printer frame with mass $m_2$ [kg] is described by Equation (2):

$${\ddot{x}}_2={\displaystyle \frac{k_2}{m_2}}{x}_1-{\displaystyle \frac{\left(k_2+k_3\right)}{m_2}}{x}_2+{\displaystyle \frac{k_3}{m_2}}{x}_3+{\displaystyle \frac{{\beta }_2}{m_2}}{\dot{x}}_1-{\displaystyle \frac{\left({\beta }_2+{\beta }_3\right)}{m_2}}{\dot{x}}_2+{\displaystyle \frac{{\beta }_3}{m_2}}{\dot{x}}_3$$

(2)

where:

• ${\ddot{x}}_2$—printer frame acceleration [m/s²];
  ${\dot{x}}_2$—printer frame velocity [m/s];
  $x_2$—printer frame displacement [m];
  $k_3$—stiffness between the printer frame and printer extruder carriage motion system [N/m];
  ${\beta }_3$—coefficient of energy dissipation between printer frame and printer extruder carriage motion system [N × s/m].

The motion of the printer extruder carriage motion system with the total mass $m_{3t}$ [kg] was described by Formula (3):

$$\begin{array}{cc}\hfill {\ddot{x}}_3={\displaystyle \frac{k_3}{m_{3t}}}{x}_2 -& {\displaystyle \frac{\left(k_3+k_4+k_5\right)}{m_{3t}}}{x}_3+{\displaystyle \frac{\left(k_4+k_5\right)}{m_{3t}}}{x}_4+{\displaystyle \frac{R{k}_4}{m_{3t}}}{\theta }_4+{\displaystyle \frac{R{k}_5}{m_{3t}}}{\theta }_5+\hfill \\ & +{\displaystyle \frac{{\beta }_3}{m_{3t}}}{\dot{x}}_2-{\displaystyle \frac{\left({\beta }_3+{\beta }_4+{\beta }_5\right)}{m_{3t}}}{\dot{x}}_3+{\displaystyle \frac{\left({\beta }_4+{\beta }_5\right)}{m_{3t}}}{\dot{x}}_4+{\displaystyle \frac{R{\beta }_4}{m_{3t}}}{\dot{\theta }}_4+{\displaystyle \frac{R{\beta }_5}{m_{3t}}}{\dot{\theta }}_5\hfill \end{array}$$

(3)

where:

• ${\ddot{x}}_3$—printer extruder carriage motion system acceleration [m/s²];
  ${\dot{x}}_3$—printer extruder carriage motion system velocity [m/s];
  $x_3$—printer extruder carriage motion system displacement [m];
  $k_4$—stiffness of belt section 1 [N/m];
  $k_5$—stiffness of belt section 2 [N/m];
  ${\beta }_4$—coefficient of energy dissipation in belt section 1 [N × s/m];
  ${\beta }_5$—coefficient of energy dissipation in belt section 2 [N × s/m];
  m₄, m₅—masses of pulleys 4 and 5 [kg];
  $R$—radius of pulleys 4 and 5 [m].

The total mass $m_{3t}$ [kg] of the printer extruder carriage motion system comprises mass $m_3$ of extruder motion system frame [kg], mass m₄ of pulley 4 [kg], mass m₅ of pulley 5 [kg], and mass of extruder carriage [kg], according to Formula (4):

$$m_{3t}=m_3+m_4+m_5+m_6$$

(4)

The motion of the printer extruder carriage with mass $m_6$ [kg] is described by Equation (5):

$$\begin{array}{cc}\hfill {\ddot{x}}_4={\displaystyle \frac{\left(k_4+k_5\right)}{m_6}}{x}_3 -& {\displaystyle \frac{\left(k_4+k_5\right)}{m_6}}{x}_4-{\displaystyle \frac{R{k}_5}{m_6}}{\theta }_5+{\displaystyle \frac{R{k}_4}{m_5}}{\theta }_4+{\displaystyle \frac{\left({\beta }_4+{\beta }_5\right)}{m_6}}{\dot{x}}_3+\hfill \\ & -{\displaystyle \frac{\left({\beta }_4+{\beta }_5\right)}{m_6}}{\dot{x}}_4-{\displaystyle \frac{R{\beta }_4}{m_6}}{\dot{\theta }}_4-{\displaystyle \frac{R{\beta }_5}{m_6}}{\dot{\theta }}_5+m_e{\omega }^{2}r\sin \omega t\hfill \end{array}$$

(5)

where:

• ${\ddot{x}}_4$—printer extruder carriage acceleration [m/s²];
  ${\dot{x}}_4$—printer extruder carriage velocity [m/s];
  $x_4$—printer extruder carriage displacement [m].

The centrifugal force resulting from the unbalance $m_e$ at the eccentricity $r$ of the fan rotor rotating with speed $\omega$ and directly loading the printer extruder carriage was calculated by Formula (6):

$$F_r=m_e{\omega }^{2}r$$

(6)

The component of the centrifugal force resulting from the unbalance of the fan rotor tangential to the horizontal displacement of the printer carriage was calculated using Formula (7):

$$F_{rx}=F_r\sin \phi =m_e{\omega }^{2}r\sin \omega t$$

(7)

where:

• $\phi$—angular displacement of the fan rotor unbalance center [rad].

The component of the centrifugal force resulting from the fan rotor imbalance, which is normal to the horizontal displacement of the printer carriage, was calculated using Formula (8):

$$F_{rz}=F_r\cos \phi =m_e{\omega }^{2}r\cos \omega t$$

(8)

The rotational motion of pulley 4 with mass $m_4$ [kg] was described by Formula (9):

$${\ddot{\theta }}_4=-{\displaystyle \frac{2k_4}{m_{4}R}}{x}_4+{\displaystyle \frac{2k_4}{m_{4}R}}{x}_3+{\displaystyle \frac{2\left(k_4+k_6\right)}{m_4}}{\theta }_4+{\displaystyle \frac{2k_6}{m_4}}{\theta }_5+{\displaystyle \frac{2{\beta }_4}{m_{4}R}}{\dot{x}}_3-{\displaystyle \frac{2{\beta }_4}{m_{4}R}}{\dot{x}}_4+-{\displaystyle \frac{2\left({\beta }_4+{\beta }_6\right)}{m_4}}{\dot{\theta }}_4+{\displaystyle \frac{2{\beta }_6}{m_4}}{\dot{\theta }}_5+{\displaystyle \frac{2T_{input}}{m_4{R}^2}}$$

(9)

where:

• ${\ddot{\theta }}_4$—pulley 4 angular acceleration [rad/s²];
  ${\dot{\theta }}_4$—pulley 4 angular velocity [rad/s];
  ${\theta }_4$—pulley 4 angular displacement [rad];
  k₆—stiffness of belt section 3 [N/m];
  ${\beta }_6$—coefficient of energy dissipation in belt section 3 [N × s/m].

The input torque T_input loading pulley 4 was determined using the velocity vs. time profile utilized as a typical printer input waveform (Figure 8) and the corresponding torque vs. time profile (

Table 1. Torque $T_{input}$ vs. time $t$ profile utilized as a typical printer input ($∆t$ = 0.00001 s—short time necessary from the point of view of the calculation stability [21], $a_{max}$ = 0.5 m/s², and $a_p$ = 0.25 m/s²).

$\mathit{t}$	0	$∆\mathit{t}$	${\mathit{t}}_1$	${\mathit{t}}_1$$+ ∆\mathit{t}$	${\mathit{t}}_2$	${\mathit{t}}_2$ $+ ∆\mathit{t}$	${\mathit{t}}_3$	${\mathit{t}}_3$ $+ ∆\mathit{t}$
$T_{input}$	0	$a_{max}{m}_{6}R$	$a_{max}{m}_{6}R$	$a_p{m}_{6}R$	$a_p{m}_{6}R$	${-a}_p{m}_{6}R$	$-a_p{m}_{6}R$	${-a}_{max}{m}_{6}R$
$t$	$t_4$	$t_4$ + $∆t$	$t_{4z}$	$t_{4z}$ + $∆t$	$t_5$	$t_5$ + $∆t$	$t_6$	$t_6$ + $∆t$
$T_{input}$	${-a}_{max}{m}_{6}R$	0	0	${-a}_{max}{m}_{6}R$	${-a}_{max}{m}_{6}R$	${-a}_p{m}_{6}R$	${-a}_p{m}_{6}R$	$a_p{m}_{6}R$
$t$	$t_7$	$t_7$ + $∆t$	$t_8$	$t_9$	$t_9$ + $∆t$	$t_{10}$	$t_{10}$ + $∆t$	$t_{11}$
$T_{input}$	$a_p{m}_{6}R$	$a_{max}{m}_{6}R$	$a_{max}{m}_{6}R$	$a_{max}{m}_{6}R$	$a_p{m}_{6}R$	$a_p{m}_{6}R$	${-a}_p{m}_{6}R$	${-a}_p{m}_{6}R$
$t$	$t_{11}$ + $∆t$	$t_{12}$	$t_{12}$ + $∆t$	$t_{12z}$	$t_{12z}$ + $∆t$	$t_{13}$	$t_{13}$ + $∆t$	$t_{14}$
$T_{input}$	${-a}_{max}{m}_{6}R$	${-a}_{max}{m}_{6}R$	0	0	${-a}_{max}{m}_{6}R$	$-a_{max}{m}_{6}R$	${-a}_p{m}_{6}R$	${-a}_p{m}_{6}R$
$t$	$t_{14}$ + $∆t$	$t_{15}$	$t_{15}$ + $∆t$	$t_{16}$
$T_{input}$	$a_p{m}_{6}R$	$a_p{m}_{6}R$	$a_{max}{m}_{6}R$	$a_{max}{m}_{6}R$

).

where:

• $a_{max}$—maximum acceleration of the printer extruder carriage [m/s²];
  $a_p$—acceleration of the printer extruder carriage [m/s²];
  0-$t_1$—maximum acceleration $a_{max}$ up to jerk speed $v_j$ $t_1={\displaystyle \frac{v_j}{a_{max}}}$;
  $t_2$—constant acceleration $a_p$ up to the desired speed $v_p$ of the printer extruder carriage $t_2=t_1+{\displaystyle \frac{v_p-v_j}{a_{max}}}$;
  $t_3$—constant deceleration ${-a}_p$ to reach jerk speed $v_j$ $t_3=t_2+\left(t_2-t_1\right)$;
  $t_4$—maximum deceleration ${-a}_{max}$ from jerk speed $v_j$ to stop $t_4=t_3+t_1$;
  $t_{4z}$—z-hop time—time for the extruder to hop to the next layer $t_{4z}=t_4+0.04s$;
  $t_5$—maximum deceleration ${-a}_{max}$ to reach negative jerk speed $v_j{t}_5=t_{4z}+t_1$;
  $t_6$—constant deceleration ${-a}_p$ from negative jerk speed $-v_j$ to negative print speed ${-v}_p$ $t_6=t_{4z}+t_2$;
  $t_7$—constant acceleration from negative print speed $-v_p$ to reach negative jerk speed $t_7=t_{4z}+t_3$;
  $t_8$—maximum acceleration from negative jerk speed to stop $t_8=t_{4z}+t_4$.
  The second cycle involves the following:
  $t_9$—maximum acceleration $a_{max}$ up to jerk velocity $v_j$ $t_9=t_8+t_1$;
  $t_{10}$—constant acceleration $a_p$ up to the desired speed $v_p$ of the printer extrude carriage $t_{10}=t_8+t_2$;
  $t_{11}$—constant deceleration ${-a}_p$ to reach jerk speed $v_j$ $t_{11}=t_8+t_3$;
  $t_{12}$—maximum deceleration ${-a}_{max}$ from jerk speed $v_j$ to stop $t_{12}=t_8+t_4$;
  $t_{12z}$—z-hop time—time for the extruder to hop to the next layer $t_{12z}=t_8+t_{4z}$;
  $t_{13}$—negative direction maximum deceleration ${-a}_{max}$ to reach negative jerk speed $v_j{t}_{13}=t_8+t_5$;
  $t_{14}$—constant deceleration ${-a}_p$ from negative jerk speed $-v_j$ to negative print speed ${-v}_p$ $t_{14}=t_8+t_6$;
  $t_{15}$—constant acceleration $a_p$ from negative print speed $-v_p$ to reach negative jerk speed $-v_j$ $t_{15}=t_8+t_7$;
  $t_{16}$—maximum acceleration $a_{max}$ from negative jerk speed $-v_j$ to stop $t_{16}=t_8+t_8$.

The rotational motion of pulley 5 with mass $m_5$ [kg] was described by Formula (10):

$${\ddot{\theta }}_5={\displaystyle \frac{2k_5}{m_{5}R}}{x}_3-{\displaystyle \frac{2k_5}{m_{5}R}}{x}_4-{\displaystyle \frac{2\left(k_5+k_6\right)}{m_5}}{\theta }_5+{\displaystyle \frac{2k_6}{m_5}}{\theta }_4+{\displaystyle \frac{2{\beta }_5}{m_{5}R}}{\dot{x}}_3-{\displaystyle \frac{2{\beta }_5}{m_{5}R}}{\dot{x}}_4+{\displaystyle \frac{2{\beta }_6}{m_5}}{\dot{\theta }}_4-{\displaystyle \frac{2\left({\beta }_5+{\beta }_6\right)}{m_5}}{\dot{\theta }}_5$$

(10)

2.4. Determination of Model Parameters

The system includes non-conservative forces defined by both linear and nonlinear parameters. The main non-conservative forces present in the system include friction, damping, and torque applied to drive the system. Some of the stiffnesses within the system, although conservative, exhibit non-linearity, such as those found at the frame–gantry interface and the drive belt stiffnesses, similar to the model used in the studies described in [21].

2.4.1. System Friction

The extruder carriage system of the printer was supported by durable rubber wheels mated with an aluminum rail. The primary source of friction in this system is the rolling resistance friction between the extruder carriage and the extruder frame rail, which is significantly less than sliding friction. Typically, the extruder carriage system is supported by a set of linear ball bearings that usually exhibit a friction coefficient μ of less than 0.005 [56] when well-maintained. The standard rolling resistance coefficient for this type of system ranges from μ = 0.002 to 0.005. The joint’s normal force arises solely from the weight of the extruder carriage, which is significantly limited by the dimensions and rigidity of the printer frame, as well as by the component centrifugal force perpendicular to the motion direction of the extruder carriage. Typically, the weight of the extruder carriage is below $F_n=10 \mathrm{N}$ under normal atmospheric conditions (i.e., the mass of the extruder carriage, its parts, and its support structure is below 1 kg). The component of the centrifugal force normal to the direction of travel of the carriage depends on the unbalanced mass, eccentricity, and rotational speed, which can reach up to 7000 rpm. For a rotor mass $m_r$ of 0.02 kg, and device unbalance class $G_{class}$ of 2.5, according to ISO 21940-11:2016 [57], the permissible eccentricity $e$ is calculated from Formula (11):

$$e={\displaystyle \frac{9549·G_{class}}{n}}$$

(11)

The corresponding unbalance U [g mm] is calculated from Formula (12):

$$U=e·m_r$$

(12)

The corresponding centrifugal force $F_r$ is calculated from Formula (13):

$$F_r=U·{\omega }^2$$

(13)

The calculated value of permissible eccentricity was 3.4 µm. Assuming that an eccentricity can increase to 10 µm due to wear of the fan rotor bearing or permanent fan rotor deflection, the unbalance $U$ can reach value of 0.2 g·mm. Centrifugal force can reach a value of 0.11 N, and its normal component $F_{rn}$ can also reach this value.

The Coriolis force, the sense of which, depending on the direction of carriage motion, can be the same or opposite to the normal force $F_{rn}$ resulting from the fan rotor unbalance, can be estimated from Formula (14):

$$F_C=2{ ·m}_r·v_{max}·\omega$$

(14)

and can reach the value of 30 N.

The friction force $T$ is estimated from Formula (15):

$$T=\mu ·\left(F_{rn}+F_C\right)$$

(15)

and can occasionally reach values of up to 0.15 N. However, for most of the operating cycle of the printer extruder carriage, they are at least twice as small. Consequently, the effect of friction between the extruder carriage and its rails has been neglected.

2.4.2. System Damping

Assuming that friction is negligible, the other non-conservative forces arise from the presence of dampers in the system. The dampers are entirely natural and originate from the printer mount, the frame, and the belts that operate the extruder carriage. In this model, the belts are considered to have lumped linear dampers with fixed damping coefficients. A comprehensive investigation by Shangguan and Zeng [58] revealed that the damping coefficient in the drive belts varied with the excitation frequency, but was independent of the belt length, confirming this assumption. The remaining three dampers in the system, specifically the workbench (${\beta }_1$), the frame (${\beta }_2$), and the interface between the frame and the extruder carriage (${\beta }_3$), are likewise assumed to be linear with fixed coefficients. This assumption for ${\beta }_1$ and ${\beta }_2$ is supported by the unchanging and stable natural conditions of these structures. In the majority of FDM machines, this interface typically includes a linear bearing moving along a linear rail of some type, or wheels attached to a rectangular aluminum alloy rail. This is a one-degree-of-freedom issue; therefore, it inherently possesses a constant damping coefficient for a given beam length. The research on structural damping carried out by Hunt et al. [59], Liu [60], and Pérez-Peña et al. [61] also corroborates this assumption.

The damping coefficients ${\beta }_j$ were estimated from Formula (16):

$${\beta }_j=\zeta ·2·\sqrt{m_j·K_j}$$

(16)

where $j$ = 1, 2, 3; $\zeta$—assumed damping ratio for Al alloy constructions, which can vary in the range 0.00001–0.001 [62,63], but can reach values up to 0.2 [64,65] (during simulation, it was assumed that the damping ratio $\zeta$ reaches value of 0.02); $m_1$, $m_2$, and $m_3$—masses of modelled movable bodies; and $K_1=k_1+k_2$, $K_2=k_2+k_3$, and $K_3=k_3$—corresponding stiffness of modelled sets of springs acting on bodies with masses $m_1$, $m_2$, and $m_3$, respectively (Figure 7).

It was also assumed that the damping coefficient ${\beta }_4$, ${\beta }_5$, and ${\beta }_6$ in sections I, II, and III of the belt driving extruder carriage (Figure 7) are independent on the belt geometry and reach constant values of 118.725, similarly to case described by [21].

2.4.3. System Stiffness

The model’s stiffnesses consist of a combination of linear and non-linear springs, which are determined by the particular FDM printer setup being analyzed. The easiest to examine are the printer base and frame stiffnesses, which are linear since they are static structures and do not noticeably alter in shape during the process. Therefore, $k_1$ and $k_2$ are constant. The stiffness of the junction between the frame and the extruder gantry ($k_3$) is greatly influenced by the specific setup of the facility, along with the placement of the connector on the rail. It can be linear or non-linear, based on the precise configuration utilized. To apply it to a particular machine setup, this connection needs to be determined. It is known that the general model will function based on $z$. Therefore, $k_3=f\left(z\right)$. In the present study, the $z$ coordinate was chosen in the middle of its range of variable values; thus, it was fixed, and the stiffness $k_3$, as well.

The stiffness $k_1$ was estimated using a simulation model based on the FEM method. The model covers the printer’s bottom plate 1 and its four attachments 2 to the rigid ground (Figure 9). For simplicity, all elements of the model were assumed to be homogeneous isotropic elastic solids made of the same AlSi alloy, for which the material model included a Young’s modulus of 71 GPa and a Poisson’s ratio of 0.33 [66].

The boundary conditions included fixing the lower surfaces (f) of the attachments that connect the plate to the ground. An FX force of 1 N was applied to the upper surface of the plate in the X direction. A mesh of finite tetrahedral elements with four nodes each was automatically generated using Autodesk Inventor 2025 software, assuming the program’s default values controlling the mesh parameters (Figure 10). Each node possessed three degrees of freedom, being displaced along three axes, u_X, u_Y, and u_Z, respectively.

The stiffness $k_2$ was also estimated using a simulation printer frame model based on the FEM method. The model covers the printer upper plate 1 and five brackets 2, attaching the upper plate to the printer’s bottom plate (Figure 11). The grid of finite elements and material model were the same as in the model of the printer bottom plate. The boundary conditions included fixing the lower surfaces of the brackets connecting the upper plate to the printer’s bottom plate. An FX force of 1 N was applied to the upper surface of the upper plate in the X direction (Figure 12).

The stiffness $k_3$ was also estimated using a simulation model of the extruder drive movement assembly based on the FEM method. The model covers the main beam 1, extruder guide strip 2, two guides 4a and 4b of the extruder drive movement assembly, and two supports for such guides 3a and 3b (Figure 13). The grid of finite elements and material model were the same as in the model of the printer bottom plate. The boundary conditions included fixing the surfaces f chosen for two guides of the extruder drive movement assembly. Such guide surfaces resist the movement of the assembly caused by the action of the force F_X. A force F_X of 1 N was applied to the upper surface of the main beam in the X direction (Figure 14).

The stiffnesses of the belt sections are the most challenging to model, since the upper two ($k_4$ and $k_5$) continuously alter their lengths during processing. The conventional model for the stiffness of belts [67] that vary in length over time is given by Formula (17):

$$k={\displaystyle \frac{C_{sp}b}{L\left(t\right)}}$$

(17)

where $C_{sp}$ represents the belt’s characteristic stiffness, $b$ denotes the width of the belt, and $L\left(t\right)$ indicates the length at time $t$. This presumes that the belts do not undergo considerable stretching while in use. The findings of the belt stiffness research [67] indicate that there is no significant stretching. Consequently, the belt’s stiffness is extremely non-linear and relies entirely on the conditions. The stiffnesses $k_4$, $k_5$, and $k_6$ are determined from Formulas (18)–(20).

$$k_4={\displaystyle \frac{C_{sp}b}{L_1+{ x}_4}}$$

(18)

$$k_5={\displaystyle \frac{C_{sp}b}{L_2-x_4}}$$

(19)

$$k_6={\displaystyle \frac{C_{sp}b}{L}}=const$$

(20)

assuming that the characteristic stiffness $C_{sp}$ and the belt width $b$ are provided and that both belt sections have the same values (either linear or non-linear). $L_1$ represents the distance of the belt from the left pulley to the extruder, $L_2$ indicates the distance from the extruder to the right pulley, and $L$ is the separation between the centers of the driver and driven pulleys. The value of $C_{sp}$ can be fairly estimated as a constant for the GT2 belts [67] found on the majority of FDM machines. The final stiffness value $k_6$ was considered constant as the belt length remains nearly unchanged during processing, and the belt width is also stable.

2.4.4. Extruder Carriage Belt Pre-Load

The impact of belt pre-load affects the initial stiffness value of the belt [58]. The standard belt pre-load $F_{pl}$ for FDM machines ranges from 35 N to 45 N, which is sufficient to prevent the belt segment from loosening during acceleration but not high enough to substantially elongate the belt. According to Wang et al. [67], the GT2 belt had an average failure tension of approximately 615 N; thus, a pre-load of 35–45 N will not greatly impact deflection. As mentioned earlier, the purpose of the pre-load is to establish the initial stiffness of the belt; it is presumed that the belt has had sufficient time to adjust after being pre-loaded before its use, ensuring that the pre-load value remains uniform across all sections of the belt. Consequently, the stiffnesses $k_4$, $k_5$, and $k_6$ are determined from Formulas (21)–(23).

$$k_4={\displaystyle \frac{C_{sp}b}{L_1+x_4}}+{\displaystyle \frac{F_{pl}}{L_0}}$$

(21)

$$k_5={\displaystyle \frac{C_{sp}b}{L_2-x_4}}+{\displaystyle \frac{F_{pl}}{L_0}}$$

(22)

$$k_6={\displaystyle \frac{C_{sp}b}{L}}+{\displaystyle \frac{F_{pl}}{L_0}}=const$$

(23)

The value of $F_{pl}/L_0$ remains constant and is determined before the commencement of the extruder carriage movement. The value of $F_{pl}$ is constant and calculated before the start of the extruder carriage motion. The value of $L_0$ was estimated to equal the length of the belt.

2.5. Thermal Analysis

The simulation model was developed to estimate the temperature distribution in the extruder carriage. In the model, components were modelled as uniform solids bonded to each other. To simplify the presence of the analysis, the fan was omitted in the model, which corresponds to less cooling conditions. The boundary conditions of the model utilized the default set of parameters including fixed adiabatic walls (for Al walls contacting with air) with an initial temperature of 30 °C, the material model corresponding to the AlSi11 cast alloy with a density of 2.79 g/cm³, the specific heat of 910 J/kgK, and the thermal conductivity of 140–170 W/mK [68]—it was assumed to be equal to 140 W/mK during calculations. Heat generation was modelled by placing on a wall mating with a heater generating a heat power of 11 W (Figure 15), assuming the condition of uniform heat generation intensity per unit of surface area per unit of time. The program automatically generated the grid of finite tetrahedral elements. Each finite element possessed four nodes, each of which had a degree of freedom in the form of temperature.

3. Results and Discussion

Solving the developed model required determining the stiffnesses of the printer elements, which could not be determined analytically. The stiffnesses k₁, k₂, and k₃ were estimated using a simulation model based on the FEM method.

3.1. Stiffness of the Components Chosen

The resulting displacement values along the X axes of the model presented in Figure 6 and Figure 7 are shown in Figure 16. Such displacements can reach a value of about 1.57 × 10⁻⁵ mm. Assuming the average value of displacement in the X-axis direction, near the center of the printer’s bottom plate equal to 0.00001 mm, the calculated value of stiffness $k_1$ was 100,000 N/mm.

The resulting displacement values along the X-axis of the printer frame model presented in Figure 10 and Figure 11 are shown in Figure 17. Such displacements can reach a value of about 0.0047 mm. Assuming the average value of displacement in the X-axis direction, near the position characterized by the middle value of the printer upper plate length along the Z-axis equal to 0.003 mm, the calculated value of stiffness $k_2$ was 333 N/mm.

The resulting displacement values along the X-axis in the model of the extruder drive movement assembly presented in Figure 12 and Figure 13 are shown in Figure 18. Such displacements can reach a value of about 1.22·10⁻⁴ mm. Assuming the average value of displacement in the X-axis direction, near the guide of the assembly, equal to 0.00003 mm, the calculated value of stiffness $k_3$ was 33,333 N/mm.

3.2. Damping Coefficients

The estimated values of damping coefficients $b_1$, $b_2$, and $b_3$ were shown in

Table 2. The estimated values of damping coefficients $b_1$, $b_2$, and $b_3.$

Damping Coefficient	Values [N/m·s]
$b_1$	120
$b_2$	70
$b_3$	25

. The damping coefficients $b_1$ were 20% higher compared to those used during the studies described in [21], while the values of damping coefficients $b_2$ and $b_3$ were much lower compared to those from [21]. This was due to differences in the geometry and type of 3D printers analyzed.

3.3. Displacements of Printer Extruder Carriage

The sample waveforms of the analyzed displacements obtained from MATLAB Simulink for the mass $m_6$ (Figure 7) equal to 0.25 kg (before the modification of the extruder head) were shown in Figure 19. It can be seen that displacements $x_1$, $x_2$, and $x_3$ were several orders lower than displacement $x_4$. Displacement $x_4$ was twice lower compared to that obtained during the studies described in [21]; however, the latter was obtained at almost twice higher accelerations. It was mainly due to the much lower values of stiffnesses $k_4$, $k_5$, and $k_6$ compared to stiffnesses $k_1$, $k_2$, and $k_3.$ The vibrational character of some periods visible in the waveforms of displacements $x_1$, $x_2$, and $x_3$ may make it difficult to control the printer extruder carriage movement to some extent. Displacements ${\theta }_4$ and ${\theta }_5$ had the same waveform in the function of time as displacement $x_4$, confirming the continuous dependence and good agreement between them.

An enlarged fragment of Figure 19a for displacement $x_4$ near its maximum value (reaching the desired speed value $v_p$ by printer extruder carriage) is shown in Figure 20. It can be seen that such a displacement rose and stabilized until the sharp drop resulting from the extruder’s duty cycle.

The sample waveforms of the analyzed displacements obtained from MATLAB Simulink for the mass $m_6$ (Figure 7) equal to 0.21 kg (after modification of the printer extruder head) without the operation of the extruder fan were shown in Figure 21. Moreover, in this case, it is visible that the displacements $x_1$, $x_2$, and $x_3$ were several orders of magnitude lower than displacement $x_4$. Displacement $x_4$ was also twice lower compared to that obtained during studies described in [21]; however, the latter was obtained at almost twice-higher accelerations. Displacements ${\theta }_4$ and ${\theta }_5$ also had the same waveform in the function of time as displacement $x_4$, confirming the continuous dependence and good agreement between them.

An enlarged fragment of Figure 21a for displacement $x_4$ near its maximum value is shown in Figure 22. The displacement raised and stabilized until the sharp drop resulted from the extruder’s duty cycle. However, a visible relatively slow increase in displacement $x_4$ is preceded by a successive rapid growth and then decrease, which is most likely due to the combined effect of stiffness and damping, as well as the changing inertias of the components on the dynamics of the extruder motion with mass $m_6$ under conditions of acceleration change. It can also be seen that decreasing the mass $m_6$ by 20% resulted in a displacement of the printer extruder carriage that is approximately 2% lower. It was probably due to the lower inertia of the extruder, especially during the decelerating phase.

The sample waveforms of the analyzed displacements obtained from MATLAB Simulink for the mass $m_6$ equal to 0.21 kg when the fan of extruder was operated were shown in Figure 23. The effect of the fan operation is more clearly seen in the waveforms of displacements $x_1$, $x_2$, and $x_3$ in the form of additional periodic changes therein.

An enlarged fragment of Figure 23a for displacement $x_4$ near its maximum value is shown in Figure 24. It is noticeable that such a displacement rose and oscillated until the sharp drop resulted from the extruder’s duty cycle. However, the rise mentioned causes displacement $x_4$ to increase by less than 1% compared to the conditions without fan operation.

3.4. Thermal Analysis of Extruder Head

The resulting temperature distribution in the bodies in the extruder carriage simulation model is shown in Figure 25. The temperature distribution in the head and heat sink assembly shows that, at a head temperature of 301.7 °C, the heat sink temperature reached approximately 78 °C. In reality, the radiator temperature can be approximately 20% lower, which is confirmed by practical tests [40]. This difference is related to the heat transfer between contact surfaces. The simulated thermal transfer uniformly distributed over the entire interface between contacting solids in reality takes place between limited, almost point-like zones, limiting the amount of heat conducted through, similar to the studies described in [69].

4. Conclusions

The proposed design of the extruder utilized the results from a numerical analysis of all its key elements and a simulation of the heat distribution from the printer head to the heat sink.

The presented design of the “Direct” extruder operating with FFF technology, with a mass not exceeding 210 g, made it extremely light, with, simultaneously, a high value of the extrusion force at a level of up to 74 N. This makes the presented design stand out in comparison to the competing solutions commonly available in the market.

The vibrational character of some periods visible in the waveforms of displacements of the components of the prototype 3D printer may make it difficult to control the printer extruder carriage movement to some extent.

The 20% decrease in the mass of the extruder head resulted in about a 2% reduction in its displacement under the applied load course. This was probably because of the lower inertia of the extruder, especially during the decelerating phase. This can be improved using the compensation elements in the control algorithm for the extruder-head-driving system.

Turning on the extruder fan increased the extruder head displacement and caused its oscillations, however, by less than 1% compared to the conditions without fan operation. This effect can be furthermore limited by the modification of the extruder head by a rotation of the extruder fun by 90 degrees in the horizontal plane.

Thermal calculations confirmed the accuracy of the design and a satisfactory temperature distribution. The presented solution allows for obtaining the stable extrusion force at the level of 15.5 N observed during the initial research of the printing process of bearing bushing for ball joints.

Considering the ball joint bushings printed on the prototype 3D printer before the modification of the extruder head, 50% of the damage occurred, while, for the ball joint bushings manufactured with the 3D printer after its modification, the damage rate was approximately 14%.