As an engineer, I love the math of how things work. It comes in handy when designing new products. I had a math teacher that said an engineer would assume that a cow was a sphere so he could do calculus integration to determine the volume. There is some truth to that. We, as engineers, simplify equations so that we can do the math easily.
There are two common linear rail systems for CNC routers. First, is simply supported and second, is fully supported. If you know the beam calculation terms you might want to include fixed support, but the routers I have seen do not give much support. The fixed beam support equation is the same except for the scaling factor. We will use the simply supported equation even though we really do not have a point load as the load is distributed within the bearings… again we will try to keep it simple just to get an idea of the maximum deflection in a router system.
The equation for simply supported beams deflection is deflection = WL3/48EI where W is the load, L is the Length, E is Young’s Modulus, and I is the inertia. You can find calculators on the internet. I found this one at engineering.com. The round support is is the third one down.
For example, if you are looking at designing/buying a router that the X rail rails are 30" long and the Y axis is 24" long. (Note the travel distances will be less maybe 24" x 18".) Also, remember there are 2 rails for the X and 2 rails for the Y.
If the rails are 3/4" in diameter and there is 20 lbs of force when we plunge the router into the work, then the maximum X rail deflection is 0.19" and the Y maximum deflection would be 0.09". This means the deflection could be as much as 0.29". This drops to 0.09" if the rail diameters are 1". If you want to assume that the rails are perfectly fixed at each end you can divide the deflection by 4, but this would be unrealistic for the routers I have seen on the market.
To make matters worse, when the spindle is close to the supports the deflection is less. This means the router will have different errors depending on where it is and how much the tool bit is loaded. This deflection is not just in the Z, but also affects the X and Y axis if we are pushing the bit.
What can be done about this deflection?
- Design/buy a router with fully supported beams.
- Use large diameter linear rods.
- Slow down and reduce the load on the bit. If the loads are small then the deflection is small.
One of the first routers I built had simply supported beams. I ended up drilling and tapping a support in the middle of the length of the beam ( In effect cutting the beam in 2 halves). Today, all my CNC routers are designed with fully supported beams.