2.77 – Week 6

This week, I finished & used my stiffness testing tool; performed preliminary error budgeting for my machine & frame; and analyzed stiffness and wear rates for my dovetail ways.


PUPS 6/Lab 6

PUPS 6/Lab 6 was an intense exercise! I tested the stiffness of my mill; modeled the stiffness & bearing wear/life in my dovetail slides; and tried to capture all of this information in my error budget.

I started out the problem set by performing some preliminary stiffness testing on my mill. The following images show my stiffness testing setups for X and Y loading (I decided not to test Z as a) it’s not a particularly sensitive direction for most machining ops, so it’s lower-priority and b) I want to prep for full system stiffness characterization before I set up to measure it):

The good news: my machine’s total stiffness is relatively similar in the X and Y directions, and it’s repeatable between tests. The bad news: it’s much lower than I had expected! I found roughly 2000 N/mm in both directions, corresponding to about 0.5 mm of deflection under 890 N (200 lbf) of load. I’m going to need to take another look at my error apportionment and see how this affects my original estimates of what performance I can expect to see (and whether I’m overestimating how much load I can reasonably expect at those performance levels). Here’s plots of X and Y stiffness:



After stiffness testing, I took my first crack at error budgeting. I initially made the following assumptions:

  • Spindle: Bearings were modeled using Alex Slocum’s bearings & shaft spreadsheet, using dimensions of current tapered roller bearings – likely to seriously underestimate stiffness since I think this assumes balls, not rollers. I’ve reached out to Timken to get approximate stiffness values at different preloads for these bearings – they will get back to me next week. I ignored translational error, although I need to revisit this – I can’t decide if we should use spindle error motion (assume spindle is moving) or just ignore it (assume spindle is static). I also used the mean dimensions of the spindle tube to generate the compliance matrix for this part – I think I’ve underestimated here as well!
  • Machine ram: The “bearings” of the machine ram are static – they’re the clamps that hold the ram in place. I found the section of the clamp with the smallest effective area after looking at both the projected area of the clamp, as well as the area of the thin section that mounts the clamp to the machine column. I analyzed the stiffness of the section under both axial and shear loading to find my linear stiffness estimates, and extrapolated from these to find torsional stiffness (although I still haven’t figured out how to deal with torsion about the Y-axis – maybe just beam bending?). For the compliance matrix, I assumed that all of the compliance in the ram came from the mounts that actually hold the spindle bearings. I modeled only the lower bearing mount, as a rectangular solid cast-iron bar.
  • X, Y, Z axes: For the X, Y and Z axes, I initially assumed that my measured system stiffness was due largely to compliance in the bearings, and that it was equally distributed between bearings. (This turns out to be a really inaccurate estimate – will get back to this later). I also assumed no compliance in the X and Y axes (since the effective “bending” section for these axes is ridiculously stubby – about 1″ long, but about 6″ x 6″ in cross section), but treated the Z axis as being a rectangular tube. I left translational & random error alone for now, although I’ll revisit later.
  • Part: Finally, for my part frame, I assumed infinitely stiff bearings, zero error motion, and zero compliance. I’ll revisit this later.

I quickly was beginning to realize that my original assumption – that the compliance of the machine was predominantly due to compliance at the bearing interfaces – was wrong: my deflection values were an order of magnitude off (7.5 mm, when they should have been 0.5 mm). After trying a few small tweaks, I began to realize that I’d have to come up with a more realistic estimate of bearing stiffness.

I then started trying to model the stiffness of the dovetail way bearing, as well as the maximum PV value & wear rate. You can see the final spreadsheet below, in my master design spreadsheet.

  • There are actually 2 different types of dovetail way in my mill: a better design (where the gib strip presses against both scraped faces in the mating way) and a worse design (where it only presses against one). I chose to model the worse design, figuring that the other would yield better performance – and I’d always prefer to underestimate performance.
  • Last Wednesday, my classmates gave me some really helpful feedback on solving this modeling problem. I haven’t captured all of their ideas yet (Hertzian contact stress for edge loading, deformation of gib screw threads), but our discussion was a good start, which I’ll keep building on as I go.
  • For actual stiffness values as a function of contact pressure, I extrapolated from a graph originally by Dolbey and Bell that I found in Precision Machine Design (pp. 429).
  • I found stiffness values that were more than 2 orders of magnitude greater than I’d estimated originally (assuming all machine stiffness was at bearings, and dividing stiffness between interfaces), even at very low screw preloads. This was true for both torsional and linear stiffnesses. I also experimented with stiffness at different preloads. Stiffness is more or less linearly related to screw torque above some baseline value: torsional stiffnesses are highest, with lateral stiffness (just resisted by the little dovetails) being lowest. Here’s a plot of stiffness as a function of preload:
  • Currently, I’m predicting that the stiffness I’ve modeled for the bearings is a significant overestimate. My model assumes that compliance is largely at the interface between the gib strip/bearing faces, wheras it’s actually much more likely to be at the interface between the screws and the gib strips (four weeny little point contacts). I may try creating a second model that tries to capture this as well, to provide a lower bound on the stiffness. It’s also a useful feature to remember when machining: I effectively have a much softer spring on one side of the dovetail way than the other, so I should probably try to set up machining ops so that they apply force through the fixed gibs (in my case, moving the part right to left against the tool in the X direction, or far to near in the Y direction). Finally, this also makes an interesting case for (someday) rebuilding the bearings with tapered gibs instead of the current screw-driven gib design.
  • Finally, I also looked into PV and wear calculations for my bearings. My expected PV value is well under my max PV value (although using the equations in FUNdaMENTALS, I don’t see how the units of these two numbers agree). My wear rate is defined as the time required to wear to a given permissible value (currently, 1 micron), and is calculated using Archard’s method. With my current assumptions – particularly that K = 0.1 e -15, corresponding to a Class 3 wear condition – I get roughly 24 hours of operation at my maximum velocity before unacceptable wear occurs. However, I’m a) not sure at all about my assumption on wear rate – would appreciate feedback on this, and b) realize that this assumes worst-case, constant full-speed operation, which isn’t realistic.

With my new stiffness measurements, I returned to update my error budget. My structural deflection values dropped from 7.5 mm to 2 mm under a 200 lbf load, most of the rest of which is due to deflection in the spindle/tool assembly (which I will revisit later). Between the other components, there was less than 0.25 mm of total deflection, which is roughly correct – I’m expecting about 0.5 mm of deflection at this load. I also updated the running parallelism of my linear ways with information that I found in a Gilman Precision catalog (spec’ing scraped linear ways as having .0005″ running parallelism over 12″). There’s still a lot of work left to do, but my error budget is on the right track.

I also thought briefly about countermeasures & risks related to my linear bearings – and specifically, regarding chip management. I seem to spend half my machining time trying to wipe chips off of the ways of my mill (particularly frustrating given that they’re typically fairly small chips!). I identified a few different solutions, including way wipers (lots of folks seem to create DIY versions of these) and chip guard bellows. I’m going to try to implement both of these, and will be sourcing cheap versions over the next 2 weeks.

Finally, I’ve started modeling my mill in CAD. I haven’t gotten very far yet, but you can see my paper notes about system measurements in my final notes below, and here’s my progress so far:


Reviewer Comments (actually – there aren’t any – we just discussed our work so far & tried to puzzle through the spreadsheet. Notes from our conversation are at the bottom)

Final Notes (includes notes from error budgeting & bearing spreadsheet development; image from in-class review session re: modeling of stiffness in bearings; paper notes re: mill measurements & CS layout)

Design Spreadsheet (includes pages with preliminary stiffness testing results; stiffness & life/wear estimation for dovetail slides)

Error Budget

Finally, there are a number of things that came up during error budgeting that I need to fix/keep in mind going forward:

  • Refine estimates of structure stiffness: My structural stiffness estimates are all likely either too low, or too high. I’d particularly like to try to improve my estimates of the stiffness of the machine column, the ram, and the knee.
  • Review bearing stiffness derivation with PREP/a professional: I’m reasonably happy with my first pass at a dovetail way analysis spreadsheet – but I’d like to review my calculations with my PREP team or one of the TAs to make sure I’m doing this right.
  • Figure out better model for spindle: Currently, the largest error in my spreadsheet is in the spindle. I need to review my model of the spindle to make its performance more accurate (and also possibly add an extra CS to separate the tool from the spindle).
  • Check assignment of section properties to spindle, ram and frame: I’m concerned about the way that I’ve assigned section properties to my spindle, ram and frame. Currently, my spindle CS is right between the bearings of the spindle, and the section I assume for this CS is the cross-section of the spindle. My ram CS is right between the clamps that hold the ram to the column, and the ram section is approximately the section that mounts the lower spindle bearing. My Z-axis CS is right at the COS of the Z-dovetail, and my Z-axis section is the bar section that I’m approximating the knee’s geometry with. Nowhere in here am I accounting for the column cross section! Which one of these assignments is done incorrectly?
  • Review assignment of bearings & frame stiffness relative to axes to make sure I’m capturing effects of moving axes correctly: Finally, I need to review the way that I assign bearing stiffness/frame stiffness relative to the different axes, to make sure that I’m capturing the motion of these axes correctly. Especially important areas for doing this are the Z and Y axes, where motion of the axis causes the length of the section of the preceding CS to change.



For Seek-And-Geek this week, I’m showcasing the stiffness tester I built to measure the axis & bearing stiffness of my mill. The idea behind this tester is to enable me to apply known loads between different points of the machine (typically the tooltip & the table), while measuring with a separate metrology system. Typically, I use a .0001″/.010″ or .001″/1 dial indicator for these tests.

My system is based around a simple concept: a screw applies a force to a spring, which presses on a load cell, which presses on a drive pin. The spring is placed between the screw and the load cell because the effective spring rate of the screw is so high: the spring takes up more displacement before it produces the maximum force the load cell can handle, giving us effectively a higher force resolution. I use a polyurethane die spring and a 1/2-20 screw – I estimate that this combination gives me an effective force resolution of 5 lbf/22 N. I also use a threaded insert to adapt between the aluminum frame of the part and the steel screw, to prevent galling.

Design Spreadsheet

Design Notes

My stiffness tester is definitely usable, but there are a few changes I’d like to make:

  • Better screw bearing: Unsurprisingly, the bearing that I built in for the rear screw – just a conical screw tip, with a conical recess on the back of the spring mount – doesn’t work. It produces too much friction, and tends to twist the spring/cell assembly as I tighten the system. I’m going to order a small section of oiled bronze, and create a new, proper bearing seat to bond into the back of the spring mount. Then, I’ll regrind and finish the screw tip.
  • Anti-rotation clamp for spring/cell: Thanks in large part to the previous problem the spring/cell assembly rotates when the screw is tightened. I’d like to machine a flat into the top of the load cell mount, and a plate with a flat stud to press against the flat in the load cell mount.
  • Return spring: Finally, I’d like to put a snap ring groove on the inside end of the dowel pin, with a low-stiffness spring between it and the inside of the frame. This will push the dowel pin back as load is reduced, keeping the load cell/spring assembly captured between it and the screw.

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