Print 3D

Edit:14 avril 2016, Cre:14 avril 2016

Engineering

Not all printer designers or users are engineers, and they not need be.
However, it might be useful to remind a few engineering stuff which can help to understand/guide design.

Thermal expansion

Printers may be installed in rooms where temperature is not constant and their own heat production change their temperature, especially if they are enclosed or semi-enclosed. Thermal expansion creates movement in mechanism and internal stress.
Material thermal expansion have a lot of effect on precision and quality of the print, so please find here values for material commonly used in printers. From the engineering toolbox

MaterialThermal expansion coefficient m/(m.°C)*10–6
Acrylic72
Aluminium22
Brass18.7
Austenitic stainless steel16
Steel12
Wood3~5

So the thermal expansion of acrylic is 6 times steel expansion and 20 times wood expansion.
That is a huge difference and explain why some people still use wood for structure, notably with heated chambers.
The difference of thermal expansion shall be handled by design as if parts are locked, the internal stress created by differential thermal expansion can skewed parts or break junction. Typically a steel rail shall be only stopped at one end and allowed to slide on the other with its supports somewhat sliding in the length axis.
Using the same material for all structure is wise, but not always easy for DIY printers.
As another example, using materials with very different coefficients in a hotend may either drive to locking parts or to leakages.

Use of the above coefficient: Expansion = coef*length*temperature difference
Please note that with unit used this makes a thermal expansion in mm for 10°C 1/100 of the coefficient for 1m length,
say for 10°C temperature difference, the thermal expansion of 1m (1000mm) of acrylic will be 0.72mm, while the expansion of wood will be 0.05mm

Stiffness.

Stiffness is key in printer design. It is the deformation of a part under a given load. It shall not be confused with resistance, which qualify the load where the part began to be destroyed. A part which is not stiff could be resistant if it does not destroy under large deformation.
Stiffness depends from the material and is generally constant for a given class of material, say resistance of steel may vary from 1 to 10 depending the quality, but its stiffness is quite constant for all nuances. So, often there is no reason to choose high quality materials if the resistance is not the main target while stiffness is the key, which is exactly the point in designing a 3D printer.

For calculation,

  • a rigidity coefficient is used for the material and another for the shape of the beam and the stiffness is the product of both.
  • A stress resistance coefficient is used for the material and another for the shape of the beam and the resistance is the product of both.

In metric units these shape coefficient are expressed:

  • Shape stiffness of a beam is generally given in mm4 or cm4 and is called inertia
  • Shape resistance capability is expressed in mm3 or cm3

as an example bending resistance of a rod 10mm diameter is 98 mm3, while its bending inertia is 491 mm4

The stiffness is given by the external part of a beam and the internal is lost material. So for a given weight, A tube will have a much higher stiffness than a rod. And large tube with thin walls are much better than smaller and thicker tubes.

Beam shapeBeam weightbending inertia
rod 12mm diameter0.88 kg/m1018 mm4
square tube 20×20×1.5 mm0.87 kg/m6300 mm4
round tube 27×1.50.85 kg/m7675 mm4

For quite the same weight, the bending stiffness varies from 1 to 7!
Not supported plain rods are a poor solution in term of stiffness and this is the reason they tend to be abandoned on large machines or with long rods as in delta.

(c) Pierre ROUZEAU
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Page mise à jour le 14/04/2016 15:09