Calculating forces on linear bearings ...

[m_c]

You've not allowed for the unequal force when cutting of centre.

Ideally, you should calculate the static load on each bearing.
Then calculate the force applied in the worst case scenario of the maximum cutting force being applied at one side of the gantry i.e. all the cutting force being applied through just 2 bearings.

In an ideal world, where the gantry is completely flex free, the cutting force would be transmitted through all 4 bearings, but in the real world it's not going to happen. Of course, some of the force will still be transmitted through the opposite side, but without lots of calculations or FEA modeling, it's simpler to use the worst case scenario.

[Jonathan]

Thanks for that. It did occur to me that the bearing forces on opposite sides of the gantry wouldn't necessarily be equal, but I wasn't sure ... so I neglected it

That makes it, for worst case, m*g/4 + F/x*(Z+z). That makes it 340N and -63N for my router which is still well within the ratings.

I'll try working out the equations for if you've got 8 bearings and 4 rails next. That's a bit more difficult though.

Another thing I've neglected is the cutting force parallel to Y (and Z if you're drilling). The former isn't going to be that great, and I think the drilling force is only significant if it's more than the weight of the gantry ...though for Z there is a moment on the bearings assuming the drill isn't centered between them.

Can anyone point me to how to work out cutting force when drilling?