Truss style Gantry

[Jonathan]

Oh look another truss.

I think you should make ends of the diagonal pieces meet, or at least close.

Lets compare this design of gantry against some of the others

Most of the gantry designs posted on this site are essentially two beams with something in between. You've added a third beam to the back and braced it, so it's bound to have better stifness, both linear and torsional, to a design with only two beams, for the same amount of material. Where you gain is you can use less material to get the required stiffness since this is an inherently more rigid structure, so you save money. Often people discount this idea due to the apparent complexity and time required to build it, but evidently that's not a problem here.

Leaving aside gravity, the gantry momentum and motor problems etc I assume all the forces that this gantry will encounter will be coming from the tip of the cutting and will be transmitted to the gantry via the four bearing blocks on the rail. (Is this right?)

Yes, let:
F_x=forces due to gantry parallel to X=force on X ballnut.
F_y=forces due to gantry parallel to Y=force on Y ballnut, total radial force on X bearings.
F_z=forces due to gantry parallel to Z=vertical component of force on X bearings.
m_g=mass of gantry
m_y=mass of Y-axis carriage
m_z=mass of Z-axis
f_cn=cutting force parallel to axis n, where n=x,y,z
an= acceleration of axis n, where n=x,y,z

Hence the forces on the gantry are:
F_x=(m_g+m_y+m_z)a_x+f_cx
F_y=(m_y+m_z)a_y+f_cy
Fz=m_za_z+f_cz+(m_g+m_y+m_z)g
There are also forces on the gantry due to the moments in each plane, but it doesn't really achieve much to list them. There's an old thread where I mentioned it here. There are some other forces involved, but they're generally negligible.

If the machine is moving forward in a straight line on along the x-axis, the forces it will generate will want to push back against the front bottom beam and pull the front top beam forward.

Yes...although it's not clear in what direction the force on the front top beam is since in addition to the force you describe from cutting, there is a force in the opposite direction due to the beam's mass, which will partially cancel. Either way there is a force on the top beam in the X direction, so having the material there to resist it is necessary.

Another assumption of mine, the distortion of steel is does not have a linear relationship to the force it is put under so if there is a small amount of force the deflection is negligible, increasing slowly until it approaches breaking point and then the deflection increases extremely quickly. (The graphs I was looking at for this seemed to be exponential or worse, is this about right?)

It depends on how the 'steel' you're analysing is supported. If you're just squashing or stretching a lump of steel, then the relationship between force and deflection is linear, like a spring (F=kx). That is until you reach the elastic limit (which clearly isn't going to happen here) and the material fails which is probably the graph you were thinking of?

If the beam is supported at both ends, then the deflection formula has a different exponent depending on the position of the force:
Euler

For twisting forces or forces parallel to the y-axis the same should apply, because the truss system isn't in one plane but is in three planes this style of truss creates a very stable form in all directions.

Agreed...

That said, I'm absolutely clueless about vibration and resonance. :(

You've said you're putting expoxy between the joints in the steel box section, presumably this is to damp vibrations in which case resonance should be less of a problem? Also note how in the .pdf I linked to earlier their analysis was based on finding the minimum number of beams to achieve the required stiffness, since by reducing the mass you increase the resonant frequency, ideally making that resonant frequency above the frequency of any forces imposed on the gantry.

Just one extra question about the gantry, for mounting SBR rails they don't have to be 'that' flat do they?

Not compared to linear guides, but what if you decide to change to linear guides at a later date? You should still aim to get it close.

Would mounting onto some cold rolled bright steel bar be accurate enough?

It depends how you mount the bar. If the bar is just bolted to the box section it will simply bend and conform to the box section, so you haven't gained anything except for maybe a smooth surface. You'd need to fix it with adjustment - for instance use lots of bolts with nuts to sandwich the steel flat bar between nuts, so you can then adjust them to make it flat. Ideally you then need an accurate reference...but for mounting SBR rails a 1m steel ruler is probably good enough.

the price for the high viscosity self leveling specialist epoxy is also very high. :(

The epoxy is cheap, i.e. west system 105, but the hardener (west system 209) isn't. If you did use it the surface would be accurate enough for either type of rail, and you wont increase the overhang as much as with other methods. The interesting bit will be making sure the surface doesn't get twisted when you fix the gantry to the X-bearings/rails, so you would ideally cast a resin surface on both.