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The technical field of the question is gantries in CoreXY 3D printers, where a gantry moves in Y direction and it holds a carriage sliding in X direction. In the following image, X is horizontal and Y is vertical.

enter image description here

When the gantry changes vertical (Y) direction suddenly, it acts as a beam supported at both ends and it deflects/resonates. This results in reduced quality (the 3D prints show waves on some surfaces) which forces more conservative settings for the acceleration and longer print times.

Different materials are commonly used for the gantry, but even if I know what moment of inertia is, I cannot grasp the (theoretical) connection between moment of inertia, density of the material, Young modulus.

A typical case is a carriage 300 g heavy mounted on one of the following profiles as stiffening support.

  • aluminium 2020 extrusions (I=7.4x10^3 mm^4, 0.25 kg, Young modulus 70 GPa)

  • square hollow aluminium profile (Young modulus 70 GPa)

    • 1 mm wall (I=4.5x10^3 mm^4, 0.1 kg)
    • 2 mm wall (I=7.8x10^3 mm^4, 0.2 kg)
  • square steel hollow profile (Young modulus 200 GPa)

    • 1 mm wall (I=4.5x10^3 mm^4, 0.27 kg)
    • 2 mm wall (I= 7.8x10^3 mm^4, 0.54 kg)

I ignored the steel rail on which the carriage actually slides, and the use of carbon fibers beams.

Which combination would result in:

  • lowest amplitude of the resonance
  • highest resonance frequency

I can see that a 2 mm alu profile has the same weight and Young's modulus as a 2020 profile, but a moment of inertia about 10 times as much: it will be better from any point of view, it's easy to understand because only one parameter changes.

However, how do I compare those performances of interest (amplitude and frequency of resonance) between steel and aluminium? for a given moment of inertia steel weight is almost 3 times as high, but so is Young's modulus.

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  • $\begingroup$ Inertia and moment of inertia and second moment of innertia are different things. Second moment of inertia is a bit like moment of innerta but in bending. Summa sumarum since both weight and crossectional shape are important, for different reasons. your much better of if you can have material further away from the bendline than having stiffer material closer to the line. This optimisation is very very case specific though. But basically you want stuff to be hollow. $\endgroup$
    – joojaa
    Jun 2, 2023 at 16:14

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Not moment of inertia, but second moment of inertia.

With regards to your last sentence of aluminum vs steel: Steel might have a modulus 3x as high, but stiffness of the structure increases to the cubed of the distance away from the bending axis. That means that with aluminum being 1/3 the density of steel, you can use a lot more volume of it for the same mass, and with all that volume the end structure can be made much stiffer than steel because you can position so much more material farther away from the bending axis.

So for same volume (i.e. aka same geometry) steel will be stiffer. For same mass. aluminum can be designed to be stiffer.

Dynamic stiffness and resonances is beyond that which I do not know but I believe it involves other material parameters such as strain-stress curves (Poisson ratio) and lossiness (viscous damping ratio?)

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  • $\begingroup$ Well its not entirely that straightforward though. It also heavily depends on what kind of stock you can get. It is easier to make very thin walled tubes out of certain materials rather than others. $\endgroup$
    – joojaa
    Jun 2, 2023 at 16:17

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