Wednesday, 24 March 2010

Bit Bender

Just some wild bendy ponderings...

How much does stuff bend ? How much support does the router table need to stay effectively true ? How thick does my MDF support beam have to be ? Is Aluminium any better ?

Calculating deformation (strain) depends on how a supporting beam is held at its ends (fixed, supported, or free). EngineersEdge list a number of useful formulas. For example, if the support beam is supported on both ends and has a single weight at its centre, the formula is y = F L^3 / ( 48 E I ), where y is the displacement height, F is the force (weight), L is the length of the beam, E is the Modulus of Elasticity, and I is the Area Moment of Inertia.

Bit Bender ©.
Use German beer for greater precision.
Das kommt vom Reinheitsgebot.

The subsequent calculation, of course, is obvious, but just in case you are exhausted from reading up on nuclear physics last night, I shall do the footwork for you:

Assume a weight of 20kg (a bit more than half the gantry weight as we use at least 2 support beams). Assume beam length L = 1m (approximate length of router table). Assume a support beam made from MDF, width 16mm and height 100mm.

As you undoubtedly know, MDF has a modulus of elasticity of E = 4 GPa = 4x10^9 N/m2 and Aluminum has E = 70GPa, ie, it bends 17.5 times less than MDF given the same volume and shape. Now the Area Moment of Inertia for a rectangular shape can be found on Wikipedia and is I = b h^3 / 12. So, for b=16mm and h=100mm, we get 1.33e-6 m4. Inserting all these values into the deflection formula, we easily find - naturally and without pen and paper: y = 0.77mm.

In layman's terms, an MDF support beam of 1 metre length, 16mm thick, and 100mm high, which is supported at both ends and holds a weight of 20kg at its centre will have a central displacement of 0.77mm.

A similar calculation can be carried out for a support beam made from a 50mm x 3mm aluminium beam. It will be displaced by 1.87mm. Things look better when both ends are fixed. MDF moves by 0.19mm and aluminium 0.47mm.

There is a difference in price: MDF = $2.92 (based on a 1200x900 board).  Aluminium = $21.98 (based on a 1000mm Aluminium angle). And a difference in weight: MDF = 1.1kg. Aluminium = 0.79 kg.

An MDF board of 76mm height and 16mm thick will bend less than 50mm x 3mm aluminium. Kind of interesting.

Based on this, it would be fine to use 100mm MDF support beams for the table.

...and I hope it's all true.


justint said...

I don't think that anyone building a hobby CNC machine has ever gorn into this much detail for selecting building materials!

justint said...

Good work though!