Lens-Free Optical Scanners for Metal Additive Manufacturing

Bibas, Charles

doi:10.1007/s11837-021-05044-8

Lens-Free Optical Scanners for Metal Additive Manufacturing

Solid Freeform Fabrication 2021
Open access
Published: 14 February 2022

Volume 74, pages 1176–1187, (2022)
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Lens-Free Optical Scanners for Metal Additive Manufacturing

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Charles Bibas ORCID: orcid.org/0000-0002-4213-3253¹

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Abstract

Galvanometer scanners (GSs) driving selective laser sintering (SLS)/selective laser melting (SLM) printers for additive manufacturing (AM) have mechanical limits. They provide inconsistent energy density across the print surface because of changes in optical path length, surface beam speed, and angle of incidence. The resulting thermal gradients may be particularly problematic for metal, whose high heat conductivity makes temperature prediction during printing critical. In this paper, we mathematically analyze and compare GSs with a new lens-free optical scanner. The results show that the latter can facilitate metal printing by providing consistent energy deposition across the print surface.

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Introduction

Galvanometer scanners (GSs) have been used for decades in SLM printing. The technology has been improved with better optomechanical control systems, better optics, better manufacturing capabilities, and even the use of machine learning.^1,2,3 These incremental improvements have significantly improved the quality of metal parts, but its inherent limitations prevent further improvements.

As the laser beam scans the print surface, fundamental factors, viz. the surface beam speed, the optical path length, and the angle of the incident beam to the surface (which affect the beam size and shape), unavoidably vary. This means that two parameters that are critical for metal printing, i.e., the energy density and surface temperature, will vary and be difficult to control.

Commonly used measures of the energy density supplied by a moving laser beam include the surface energy density or Andrew number^4,5

$$ {\text{Ed}} = P_{{\text{l}}} /\left( {v \cdot {\text{Def}}} \right) $$

(1)

where P_l is laser power, v is scan speed, and Def is laser beam diameter or hatch width, and the volumetric energy density (Ev), which extends the Andrew formula to the volume of the top layer by including the layer thickness, L_h:

$$ {\text{Ev}} = P_{{\text{l}}} /\left( {v \cdot {\text{Def}} \cdot L_{h} } \right) $$

(2)

As is clear from these equations, if the beam speed, size, and shape vary across the print surface, the energy density will also vary and be difficult to control. The resulting thermal gradients are particularly problematic for metal, whose high heat conductivity makes temperature prediction during printing critical. Numerous studies such as those of Goodridge⁴ and Bertoli⁶ suggest that Ev must remain within a narrowly defined range to maintain uniform melting of the powder. Nonuniform temperature will cause thermal stress to appear in the model,⁷ resulting in tensile residual stress on the surface and acceleration of crack propagation.

A new lens-free optical scanner (LFOS) has been developed by Tecnica to largely eliminate the current shortfalls of GS devices. The Øgon scanner maintains constant surface beam speed, constant optical path length, and an incident beam that is always perpendicular to the surface. It thus provides uniform energy density and uniform temperature across the print surface.

While many models have been developed to analyze and control the energy density and temperature distribution in AM metal printing, including those in Refs. 8,9,10,11, most consider the idealized case in which the energy input from the scanner is constant; they do not account for the inconsistency of GSs across the print surface. The Øgon, on the other hand, provides the uniform energy input that these models assume.

In this paper, we provide some mathematical analysis of the LFOS. The results show that it can facilitate metal printing by providing consistent energy deposition across the print surface.

Methods

The GS system is analyzed using the postobjective scanning method in which the scanner is positioned after the objective lens. The rays converge to be focused at the work surface even though the beam source is collimated. Choosing this method can expose weaknesses in the optical system in comparison with the preobjective method.

Although herein we analyze the GS and compare it to the Øgon, polygonal mirrors offer similar results to GS in that they deliver nonconstant values of OPL and Θ as a function of beam location, thus the temperature prediction chart is similar to that obtained using GS.

The optomechanical properties of GS and Øgon scanners are analyzed mathematically and compared with regard to the main factors complicating temperature prediction, which is critical in AM of metal: beam speed, diameter, and shape, which affect Ev, and the beam thermal gradient.

Mathematical models are used to produce a system capability chart for each scanning system using which the pros and cons of each scanner are illustrated and compared.

Galvanometer: Advantages and Limitations

Transfer functions represent the properties of the beam as an output, where the input is the command parameter(s). For the GS system, the transfer functions are a function of θ_x and θ_y (input), where $\theta_{x} = {2} \cdot \omega \cdot t$ and $\theta_{y} = {2} \cdot \omega \cdot t$, with $\omega \cdot t$ being the angular rotation of the respective motor (Fig. 1).

Position

The beam location can be described as¹²

$$ y\left( {\theta_{y} } \right) = d \cdot \tan \left( {\theta_{y} } \right) $$

(3)

$$ x\left( {y,\theta_{x} } \right) = \left( {\sqrt { d^{2} + y^{2} } + e} \right) \cdot \tan \left( {\theta_{x} } \right) $$

(4)

$$ x\left( {\theta_{x} ,\theta_{y} } \right) = \left( {d \cdot \sqrt {1 + \tan^{2} \left( {\theta_{y} } \right)} + e} \right) \cdot \tan \left( {\theta_{x} } \right) $$

(5)

and the speed as^12,13

$$ Vx\left( {\theta_{x} } \right) = \frac{{{\text{d}}x}}{{{\text{d}}\theta_{x} }} = \left( {d \cdot \sqrt {1 + \tan^{2} \left( {\theta_{y} } \right)} + e} \right) \cdot 1/\cos^{2} \left( {\theta_{x} } \right) $$

(6)

$$ Vy\left( {\theta_{y} } \right) = \frac{{{\text{d}}y}}{{{\text{d}}\theta_{y} }} = d \cdot \frac{1}{{\cos^{2} \left( {\theta_{y} } \right)}} $$

(7)

$$ V = \sqrt {\frac{{ \left( {d \cdot \sqrt {1 + \tan^{2} \left( {\theta_{y} } \right)} + e} \right)^{2} }}{{\cos^{4} \left( {\theta_{x} } \right)}} + \frac{{d^{2} }}{{\cos^{4} \left( {\theta_{y} } \right)}}} $$

(8)

Optical Path Length

The OPL is the path length from the beam source or a reference point on the optical axis to the incident point on the work surface, expressed as a function of θ_x and θ_y:¹³

$$ {\text{OPL = }}\left( {d \cdot \sqrt {1 + \tan^{2} \left( {\theta_{y} } \right)} + e} \right) \cdot \sqrt { 1 + \tan^{2} \left( {\theta_{x} } \right)} $$

(9)

Angle of Incidence

Θ, the angle of incidence, is the angle between the incident beam and the normal to the surface. It is calculated from the triangle P Px R as

$$ \Theta = \arccos \frac{d}{{\sqrt { x^{2} + \left( {\sqrt {d^{2} + y^{2} } + e} \right)^{2} } - e/\cos \left( {\theta_{x} } \right)}} $$

(10)

Alternatively, it can be expressed as a function of (θ_x, θ_y) as

$$ \Theta = \arccos \frac{1}{{\left[ {\sqrt {1 + \tan^{2} \left( {\theta_{y} } \right)} + \frac{e}{d}} \right] \cdot \sqrt {1 + \tan^{2} \left( {\theta_{x} } \right)} - \frac{e}{{d \cdot \cos \left( {\theta_{x} } \right) }}}} $$

(11)

Equations 9 and 11 shows that Θ changes across the scan area. Θ is also one of the parameters affecting the beam intensity.

The OPL is a product of two nonlinear functions that change quickly as the beam moves away from the origin (print center). The OPL is also one of the parameters affecting the beam’s focus and therefore its intensity.

Symmetry

It is evident from Eq. 3 that there is symmetry about the origin for GSy, where y(− θ_y) = − y(θ_y). Similarly, from Eq. 4, GSx is symmetrical when θ_y is held constant where x(− θ_x, θ₀) = − x(θ_x, θ₀ ) for θ_y = θ₀. Note also that x is a function of the two inputs θ_x and θ_y. However, because the two GSs are positioned at different distances from the work surface (as the distance between GSx and GSy is e), this introduces asymmetry between the two axes.

Equations (3) and (4) show that

$$ y(\theta_{0} ) \ne x(\theta_{0} )\quad {\text{for}}\quad \theta_{y} = \theta_{x} = \theta_{0} $$

Additionally, it is evident from Eqs. 6 and 7 that

$$ V_{x} (\theta_{0} ) \ne V_{y} (\theta_{0} )\quad {\text{for}}\quad \theta_{y} = \theta_{x} = \theta_{0} $$

This asymmetry and nonlinear speed make it more difficult for the two-GS system controller to maintain the constant speed required to keep Ev within the narrow target range.

Beam Size and Shape

Consider a circular beam with radius r₀ and at focus at the origin; as the beam moves across the work surface, the beam size and shape will change as Θ and OPL change.

As the beam traverses the surface and Θ deviates from zero, the beam’s image on the surface stretches from a circle to an ellipse, whose vertical and horizontal axes we denote by r_v and r_h, where v and h are the local coordinates of the image lying on the work surface. Beam augmentation is analyzed in two stages, wherein stage 1 is the effect due to OPL and stage 2 is the effect due to Θ. The two stages can be quantified as follows:

The beam radius change as a result of OPL (stage 1)¹³ is

$$ r1 = r0\left( {\left( {d \cdot \sqrt {1 + \tan^{2} (\theta_{y} )} + e} \right) \cdot \sqrt {1 + \tan^{2} (\theta_{x} )} - (d + e)} \right) \cdot {\text{Ha}} $$

(12)

The beam cross-section grows symmetrically along both local axes, resulting in a larger circular beam diameter.

The beam radius change as a result of Θ (stage 2)¹³ (where only the local vertical axis changes), can be expressed as

$$ r_{2v} = r_{0} \cdot \frac{1}{\cos (\Theta )} $$

(13)

$$ r_{2v} = \left( {\left( {\sqrt {1 + \tan^{2} (\theta_{y} )} + e/d} \right) \cdot \sqrt {\tan^{2} (\theta_{x} ) + 1} - e/(d\cos (\theta_{x} ))} \right) \cdot \left( {r0 + \left( {\left( {d \cdot \sqrt {1 + \tan^{2} (\theta_{y} )} + e} \right) \cdot \sqrt {1 + \tan^{2} (\theta_{x} )} - (d + e)} \right) \cdot Ha} \right) $$

(14)

$$ r_{2h} = r_{1} $$

(15)

$$ A = \pi \cdot r_{2h} \cdot r_{2v} = \pi \cdot r_{1}^{2} \cdot \frac{1}{\cos \left( \Theta \right)} $$

(16)

substituting 1/cos(Θ) with Eqs. 10 and 11 yields

$$ A = \pi \cdot \left( {r0 + \left( {\left( {d \cdot \sqrt {1 + \tan^{2} (\theta_{y} )} + e} \right) \cdot \sqrt {1 + \tan^{2} (\theta_{x} )} - (d + e) \cdot {\text{Ha}}} \right)} \right)^{2} \cdot \left( {\left( {\sqrt {1 + \tan^{2} (\theta_{y} )} + \frac{e}{d}} \right) \cdot \sqrt {1 + \tan^{2} (\theta_{x} )} - \frac{e}{{d \cdot \cos (\theta_{x} )}}} \right) $$

(17)

It is apparent from Eqs. 16 and 17 that the energy density E/A changes as the beam moves across the work surface. Moreover, Eqs. 16 and 17 is not symmetrical with respect to θ_x and θ_y, and for x, y coordinates:

$$ \begin{aligned} y\left( {\theta_{x} } \right) & \ne x\left( {\theta_{y} } \right) \\ A\left( {x,y} \right) & \ne A\left( {y,x} \right) \\ \end{aligned} $$

Most importantly, the tan(θ_x) and tan(θ_y) components change nonlinearly as the angles grow. This causes sharper growth in A as θ_x and θ_y grow. The asymmetry in A augmentation needs to be considered for each pixel. Therefore, the modulator needs to be configured to take into account these nonlinearities and asymmetry. This also means that part positioning on the x–y bed matters.

Furthermore, the speed transfer function is not symmetrical: V_x(θ₀) ≠ V_y(θ₀) for θ_x = θ_y = θ₀. This further highlights the asymmetry in energy deposition as the beam speed is one of the parameters dictating the energy density absorbed by the surface.

Linearity

It is apparent from Eqs. 3 and 4 that the transfer function for either x or y is nonlinear. The use is limited to small angles where tan(θ_x) or tan(θ_y) is close to linear. It is evident from Eqs. 10 and 11 that e, the distance between GSx and GSy, must be kept as small as possible in relation to d, the distance between GSy and the work surface. The smaller the value of e/d, the smaller the asymmetry around all axes.

Modulation

Modulation in a GS system is performed using a closed loop. The challenges lie in addressing nonlinear transfer functions (position, speed, and energy deposition for printing) and the mechanical inertia of the oscillatory element in the GS. The nonlinear positioning can be addressed by geometric correction techniques.¹⁴ Similarly, Zhang¹⁵ further measured data spots and performed online calibration of the GS.

Duma¹⁶ further explored the GS mechanical inertia of the oscillatory element limitation. He qualifies the modulation speed limit to be around 1.5 kHz and selected the preferred/optimized modulation waveform to be a sawtooth wave with fast and slow-moving GS where the raster method is used.

Offset/Drift Errors

Offset/drift errors occur when the GS deviates from the origin and as a result moves all respective coordinates. Offset/drift errors are temperature sensitive and must be watched closely. Offset/drift errors can be addressed by online correction on the fly, in a similar method to Zhang.¹⁵

Offset errors are measured in microradians. A typical value from a high-quality GS manufacturer is less than 10 microradians. Some manufacturers, unfortunately, amplify the offset/drift error when they make design choices to reduce Θ-related errors. As an example, a company may enlarge d (the distance from the GSy to the work surface) from 500 mm to 2000 mm. Such enlargement of d will reduce the maximum Θ required for the same size work surface, thereby reducing Θ-related errors, but will also amplify offset/drift errors.

Consider a 0.1 mm beam diameter using a hash (grid) of 0.1 mm. The position is now dictated by a longer arm d. A typical 5 µrad offset/drift error will result in a 10 µrad error in the correspondent GS θ.

If the intended location is x₀, then the new location is

$$ x_{0} \pm {\text{Err}} = x_{0} \pm d \cdot 10\;\upmu {\text{rad}} = x_{0} \pm 20\;\upmu {\text{m}} $$

This is already a change of 20% of the grid size. Temperature changes will amplify this error by a factor proportional to the temperature change. As an example, Cambridge Technology for their GS model 83xxK spec reports a zero drift parameter of 5 µrad/°C. A change of 7 °C will thus result in a position change (for d = 2000 mm) of

$$ x_{0} \pm {\text{Err}} = x_{0} \pm 5 \cdot 2 \cdot 10^{6} \cdot 7\;\upmu {\text{rad}} = x_{0} \pm 70\;\upmu {\text{m}} $$

Note also that this error is only for one GS. Adding the second GS vector will result in a total grid shift of $\pm {\text{sqrt}}(2) \cdot 70\;\upmu {\text{m}} \to \pm 99\;\upmu {\text{m}}.$

Repeatability

The repeatability error is measured in units of microradians. It is doubled when reflecting from the GS mirror. Additionally, the error is further augmented when GSx and GSy errors are combined. A typical value for one GS is 5 µrad to 10 µrad.

Energy Density

In this section, we calculate the volumetric energy density Ev (Eq. 2) as a function of the angle of incidence Θ, taking into account the changes in beam speed and beam diameter discussed above, as well as Lambert’s cosine law.

Define OPL’ as the optical path length from the GSy axis to the work surface, where d is the beam distance at focus for a beam converging in Ha. In this way, we can simplify the calculations without using θ_y or θ_x:

$$ {\text{OPL}}^{\prime } = d/\cos (\Theta ) $$

(18)

Ha can be expressed with M the beam size at the mirror and d (Fig. 2):

Since ${\text{Def}}_{0} \ll M$, we can simplify the term for M as

$$ M = {\text{Def}}_{0} + 2 \cdot d \cdot {\text{Ha}} \to {\text{Ha}} = M/\left( {2 \cdot d} \right) $$

(19)

Considering stage 1 beam enlargement and substituting Ha, we have

$$ {\text{Def}}_{1h} = {\text{Def}}_{1v} = {\text{Def}}_{1} = {\text{Def}}_{0} + 2 \cdot ({\text{OPL}}^{\prime } - d) \cdot M/\left( {2 \cdot d} \right) $$

(20)

$$ {\text{Def}}_{1} = {\text{Def}}_{0} + \left( {1/\cos \left( \Theta \right) - 1} \right) \cdot M $$

(21)

Stage 2 beam enlargement affects only the vertical axis, thus

$$ {\text{Def}}_{2v} = {\text{Def}}_{1} /\cos \left( \Theta \right) $$

(22)

where Def_2h remains unchanged. We can now calculate a new Def for the two stages combined by calculating the area of the ellipse, then normalizing it to an equal circular area, resulting in

$$ {\text{Def}} = \frac{1}{{\sqrt {\cos \left( \Theta \right)} }} \cdot {\text{Def}}_{1} = \frac{1}{{\sqrt {\cos \left( \Theta \right)} }} \cdot \left( {{\text{Def}}_{0} + \left( {\frac{1}{\cos \left( \Theta \right)} - 1} \right) \cdot M } \right) $$

(23)

The beam speed is extracted from Fig. 2 as

$$ v = {\text{d}}h/{\text{d}}\Theta = \frac{{d \left( {d \cdot \tan \left( \Theta \right) } \right)}}{d\Theta } = \frac{d}{{{\text{cos}}^{2} \left( \Theta \right)}} $$

(24)

Using the Ev in Eqs. 2, 22 and 23 and applying Lambert’s cosine law, this yields (Fig. 3)

$$ {\text{Ev}} = \frac{{P_{{\text{l}}} }}{{v \cdot {\text{Def}} \cdot L_{h} }} = \frac{{P_{{\text{l}}} \cdot \cos (\Theta )}}{{\frac{2 \cdot \pi \cdot R \cdot f \cdot d}{{\cos^{2} (\Theta )}} \cdot {\text{Def}} \cdot L_{h} }} = P_{{\text{l}}} \cdot \frac{{\cos^{3} (\Theta )}}{{L_{h} \cdot v_{0} \cdot \left( {\frac{{{\text{Def}}_{0} + \left( {\frac{1}{\cos (\Theta )} - 1} \right) \cdot M}}{{\sqrt {\cos (\Theta )} }}} \right)}} $$

(25)

$$ {\text{Ev}} = \frac{{P_{{\text{l}}} }}{{L_{h} \cdot V_{0} }} \cdot \frac{{\cos^{3} (\Theta ) \cdot \sqrt {\cos (\Theta )} }}{{\left( {{\text{Def}}_{0} + \left( {\frac{1}{\cos (\Theta )} - 1} \right) \cdot M} \right)}} $$

(26)

Temperature Prediction

The energy deposition across the work area depends mostly on Θ and the print path. Equations 25 and 26 shows that the energy deposition for each voxel depends on the voxel location, so the energy deposition varies as the beam moves across the print surface. Consider the case where the beam is moving away from the origin from voxel n to voxel (n + 1); the elongation of the beam shape on the surface means that, while the beam is activated for voxel n, voxel (n + 1) will be preheated. In contrast, when the beam is moving towards the origin, the next voxel is not preheated. This asymmetry further contributes to the nonuniform temperature distribution. Mirkoohi⁷ presented a model for temperature prediction where the laser source is normal to the print surface, moving in a straight line, emitting constant power, and moving with constant velocity. Mirkoohi solved the model for the temperature gradient at any x, y location, resulting in Eq. 27, where the model captures the melt pool geometry for x, y. In her research, Mirakoohi also uses a numerical analysis model where both models are compared with experimental data.

Equation (27) further confirms that T₀, the initial voxel temperature, is one of the parameters affecting the gradient temperature. Therefore, a higher T₀ will introduce a larger temperature gradient due to the post heat when the beam moves away from the center compared with a voxel in the same location when the beam moves towards the center.

For the temperature prediction equation to obtain T (the temperature gradient) at a distance R from the laser spot where $R = \sqrt {x^{2} + y^{2} }$, Mirkoohi⁷ (for a laser moving along the x-axis) uses

$$ T = \frac{{P_{{\text{l}}} }}{4\pi KR}\exp \left( {\frac{ - v(R - x)}{{2 \propto }}} \right) + T_{0} $$

(27)

where P_l is the laser power, v is the laser velocity, K is thermal conductivity, and α is thermal diffusivity.

Attempts to Fix the Transfer Functions of the GS in AM

An f-theta lens is needed to keep the laser beam in focus on the flat plane of the work surface and to compensate for the speed variations across the surface. The speed is kept constant by changing the laser beam angle so that the beam distance traveled along the planar work surface is linearly proportional to the input rotation angle of the beam. More specifically, the f-theta lens transforms the beam direction from the entry angle θ to an exit angle arctan(θ).

To also correct the angle of incidence and maintain a circular beam spot shape, a telecentric f-theta lens is required, and it must be at least as large as the work surface.

f-theta lenses are not perfect solutions and are not scalable. They are expensive, difficult to manufacture, have errors due to manufacturing tolerances, and cannot in practice be manufactured beyond a size limit.

Conclusions for GS

Although the GS provides 2D scanning in the x and y dimensions. the asymmetry and nonlinearity in positioning and speed make it more challenging to modulate when high-resolution, high-quality print/scanning is needed. This is especially true for AM, where the analog intensity level is important, in contrast to sensing.

Although the modulation of the y coordinate is dependent only on the θ_y, modulation, the x coordinate depends on both θ_x and θ_y. Therefore, the x modulator needs additional input. This puts a burden on the modulator when rendering a vector because the print path must be calculated to produce the outputs for each GS. Additionally, the beam cross-section changes with Θ, resulting in additional degradation of the absorbed energy at the surface. Attempts have been made to mitigate the GS positioning by Xianyu¹⁴ and the drift in situ by Zhang.¹⁵

The Mirkoohi model⁷ is based on a beam normal to the surface and constant beam speed, beam shape, and size, where the substrate is at the same temperature and the beam follows a straight-line path. Even if the GS uses a straight-line print path, the input parameters to the model will deviate from the Mirkoohi model, suggesting a more complex model where the heat flux emitted by the laser is broken down into two components: a vertical component and a horizontal component, both of which change in size and direction as Θ changes.

Additionally, the GS beam incident will vary by voxel location. Moreover, the beam is not normal to the surface, so melt pool models, such as that of Mirkoohi,⁷ do not fully reflect the GS operation, Therefore, the melt pool model is difficult to predict.

The heat deposition will vary for each voxel because of the numerous variables. Overall, the sum of all the voxels’ temperature contributes to the layer temperature prediction.

LFOS: Advantages and Limitations

The Øgon LFOS consists of two reflectors (M1 and M2; Figs. 4, 5, 6), where M1 is a rotating reflector and M2 is an annular stationary reflector. The beam source strikes M1 along the optical axis, which is also the actuator rotational axis. The beam then travels towards M2, which diverts the beam to the work surface.

Because M2 is a circular ring of radius R, it imposes a focus at a distance d = R/2 for a sphere and R for a conical ring on the x dimension of the beam (Fig. 5). The y component of the beam is focused by the M1 y surface component, and the M2 profile with the focus is selected to match the x dimension distance at R/2 from M2.

The LFOS functions performed on the beam source are:

1.
Beam steering: first, it is steered by M1 from the beam source optical axis, then by M2 back parallel to the beam source optical axis;
2.
Beam revolution around the beam source: the output beam revolves around the beam source optical axis;
3.
Beam rotation around its local optical axis;
4.
Beam focusing: the beam is fully focused in both directions at the worksurface.

The LFOS focuses a diverging beam without the use of lenses. As an example, a beam source from a multimode fiber with 105 µm cladding diameter and NA0.15 focuses back to the beam source size in the Øgon first prototype, where r = 125 mm and d = 60 mm.

Because the beam rotates around its own axis, it radiates more consistent energy.

Scanning Method

LFOS uses a modified raster scanning method in which the x-axis is scanned in arcs instead of horizontal lines. After each arc, a linear conveyor moves the LFOS a fixed vertical distance along the y-axis to scan the next arc. This method resembles polygonal mirror (PM) two-dimensional (2D) printing. However, the result is greatly superior to the PM method because the PM must print on a cylindrical conveyor to keep the focus, while the LFOS is at focus printing on a flat surface. Additionally, the slow-moving conveyor wear and tear is very small compared, for example, with a CNC actuator because there is very little travel time during the print.

Maximum Scanning Speed

The rotational speed at the work surface is controlled by the motor rotational speed (rotations per second f). When M1 is carried by a brushless motor, f = 2000 Hz (2000 rotations per second). When R = 125 mm is selected, the surface print speed is

$$ 2 \cdot \pi \cdot R \cdot f = 1570\;{\text{m}}/{\text{s}} $$

For comparison, the maximum print speed of a GS is below 10 m/s.