For a larger build volume, what lengths of 2020 aluminium do I need?

Question

TL;DR

How do I upscale a Wilson II? What lengths of aluminium¹ do I need in order to achieve a particular (increased/reduced) build volume?

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The design of the Wilson II is scalable (source: RepRapWiki - Category:Wilson):

...the design has a parametric build area, meaning it is relatively easy to scale the X, Y, and Z axis within reason.

What does parametric mean exactly, in this scenario? How does one scale up from 200x300x200¹? Also, how would that affect the Repetier/Marlin firmware?

Is it simply a matter of maintaining the ratios of the lengths of the X, Y and Z axes constant, or can the ratios be ignored? Is there a formula, or set of formulae, for this?

Has anyone gone beyond the 200x300x200 build volume? I have seen the Scalar M and XL series printers (with the XL having a print volume of 400x300x300) which, while they are not based on the Wilson, also boast of scalability:

Scalar Family 3D printers are "scalable" printers. Reviewing the idea of a reprap printer, a printer that can auto replicate and scale, we wanted to propose a 3D printer with plastic parts for you to print, and with a way to "scale" easily.

Can one (within reason) arbitrarily section various (supersized) length for the three axes and then modify the firmware accordingly, or is there a set of rules which govern the relationship between the lengths of the three axes?

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A simplistic view

As an example, the lengths (in mm) of the 2020 aluminium corresponding to the build volume of 200x300x200 are 330, 500, 400 for the X, Y, and Z axes, respectively.

Obviously, there are some constants to consider for the stepper housings, and idlers, for example. So, assuming that for X, Y and Z axes respectively, the constants are:

• 330 - 200 = 130 mm
• 500 - 300 = 200 mm
• 400 - 200 = 200 mm

If I wanted a build volume of, let's say, 400x500x300 (XYZ), would the new XYZ lengths of 2020 aluminium simply become (by adding the respective constants):

• 400 + 130 = 530 mm
• 500 + 200 = 700 mm
• 300 + 200 = 500 mm

or is there more to it than that?

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¹ I appreciate that 3030, en lieu of 2020, extruded aluminium may be required to maintain rigidity for larger print volumes

Answer 1

Long story short: ratios can be ignored. You only need to let your firmware know actual size, if it needs so. However, using golden ratio seems quite reasonable.

When talking about rigidity, keeping it will become a real issue at some point and may not be always effectively resolved by merely changing 2020 to 3030. Using 4020 or double 2020 could be better solution to compensate the most common forces during printing.