Diaphragm Flexure Testing — Part 1

Overview

The goal here is to test and compare a few different printed diaphragm flexure designs. This is a continuation of the train of thought from this post.

The comparison will be primarily based on measurements of three stiffnesses:

• Axial stiffness
• Radial stiffness
• Yaw stiffness

In a sense, the ratios of Radial to Axial and Yaw to Axial are kind of the measures of how ‘good’ the design is (for my purposes). I want very high Radial and Yaw stiffness with low axial stiffness. This combination will ensure that the flexure gives repeatable, single degree of freedom motion along that axis of compliance (so to speak).

Test Design

The plan is to use a common frame design with an open annulus for the diaphram flexure elements. The Frame and Hub will be a common design to allow for consistent mounting and loading between tests.

For radial and yaw stiffness measurements, my plan is to mount the flexure under test by the hub. I’ll then apply a load along the centerline and measure displacement at 180 degrees offset. For Yaw, I will apply the load at the corner (as shown below as F_Yaw) and measure displacement on the face directly opposite (basically load and sensor both translate along their respective edges). This is why I’ve opted for a square outer perimeter, it should enable this test approach pretty easily.

Here’s what I’m thinking for the main parameters (in mm):

That will give 40 mm of radial space for the flexure elements, which I think should be sufficient.

As I’m thinking about this a bit more, it occurs to me I’m probably going to need to make these in pairs. I want to be able to print the flexures flat on the build plate, but the flexure elements will likely only be a couple of millimeters thick. That will force me to have the frame asymmetric about the flexure, which would make it very difficult to apply a dead load to the frame through the centerline of the flexure.

Although it will take more material and assembly, if I make the flexures in pairs it will

1. Create a plane of symmetry between the two flexures along which to place the radial and yaw loads
2. Better represent the actual functional design (the end application will also use paired diaphrams)
3. Resist Pitch and Roll error motions

I was going to cut Vees in the ‘top’ surface and then place the cylindrical test masses on their sides. But I realized that’s not really going to work. Those masses only go up to like 200g, or maybe I have one 500g, but either way they’d have to be stacked. And that’s not gonna happen on top of this thing with a couple of vees. So I instead opted for three hooks on the bottom that I can run string through. I can then use my old favorite of a water bottle. I decided to add the second yaw test location because symmetry…and because it will allow me to test both postive and negative yaw loads. Since some of the concepts I have in mind are not rotationally symmetric, this will let me test how that asymmetry shows up in the stiffness.

So here’s how my ‘base bodies’ are looking.

Here is the layout for Test concept 1 (I decided to make a full test article to use for the design of the test jigs).

Here’s a look at the axial stiffness test config. The displacement sensor is looking at the flat surface on the hub (unfortunately it is off axis, but I settled).

A mass will be suspended from a string….

(aside, but big thanks to the maker behind these test mass models, they’re great!)

…the other end of that string will be anchored to a truncated steel ball that will rest in the chamfer in the center of the hub…. I screwed up the size of the truncated balls I had on hand before printing my first set of flexures, and decided to just live with it. As a result, my truncated balls don’t contact the chamfer surface, but instead are contacting the sharp edge between the chamfer and the hole. This is almost certainly going to cause contact stresses enough to deform that sharp edge, but since I’m not measuring off of the mass stack for the displacement measurement, I don’t really care if the mass is creeping at this interface. It should just deform until the contact patch is large enough to support the load and stabilize. So I convinced myself I’m not worried about it….also, I’m ‘anchoring’ to the truncated ball by tying the string to a hex nut that will rest against the bore in the ball. So clearly I’m ok with some variability at this interface :)

One thing I’m curious to check out with this setup is a comparison of Fusion’s sim predictions to measured stiffness. For a number of reasons, I don’t expect the Fusion predictions to be accurate (e.g. I’m using the automatically generated mesh, I don’t have a material definition for the actual filament, the material properties I do have don’t have properties defined for nonlinear analysis….plus I’m just not an FEA expert :) ). But I’m hoping I’ll be able to find a strong enough correlation to reality that I can use them for some back of the envelope estimating, maybe with a scaling factor or something.

So the plan is to run sims at each of the load values I will physically measure and repeat this for each design concept. For now, there are four concepts I’m going to test, along with one ‘control’. The purpose of the control is to be able to measure the deflection of the test setup itself. Because the test parts are being held by a cantilevered bit of printed plastic, I anticipate this error source to be more than negligible and I want to be able to correct for it. For the control, I went with the same outer geometry, but it is a single component instead of an assembly. The hub is rigidly fixed to the outer frame. Even though I don’t expect meaningful deflection, I’ll still go ahead and simulate this one along with the others.

The main source of error that I’m worried about is the opening up of the C structure depicted in orange below. The bottom of that C is the cantilever I mentioned, and it is directly on the primary load path for the test load, while the top of the C (the sensor mount) isn’t. So I would expect some relative motion between these two, and that relative motion is pure measurement error. Luckily, this deformation should be repeatably, regardless of the design of the part under test (as long as the axis of the applied load is roughly in the same location).

The four actual test assemblies consist of two main design variants. One version has a line of symmetry:

And the other one doesn’t:

And the other two version are the same flexure design as those shown above, but with the leaf springs being 7mm wide, instead of the 5mm width of the above.

Axial Stiffness Results

I know the below are out of order, but I’m writing them in the order I did them…and I did them in a random order…deal with it. Also, please note, all of the below are without correction from the control test….since I still haven’t printed it at the time of writing this…again, deal with it :)

Concept 1 — Symmetric 5mm

Here’s how the data looks for the axial measurements on Concept 1. Technically, the below plot is showing compliance, not stiffness, with displacement on the vertical axis and load along the horizontal. The Fusion simulations predict a significantly stiffer flexure (64% stiffer) than what is actually measured. This is in line with what I would have expected.

In general, I typically expect this sort of ‘easy’ FEA to predict a flexure to be too stiff. The main reason for this lies in the meshing. When creating these automatically generated meshes, you usually get something that looks like this.

It’s made up of a whole bunch of these tetrahedrons, a solid made up of all triangular faces.

The nice thing about these elements is that they are pretty easy to pack together to represent a complex geometry, like my flexures. But one big downside is that they have a tendency to be artificially stiff, especially when there are only a couple of elements across the body, like here within the leaf flexures themselves.

In addition to this source of extra stiffness, I assume a second significant source of error here comes from the difference between the simulation of bulk material with the reality of a 3d printed material. I don’t have any easy way of confirming this assumption (without having some sort of machined version added to the mix), but I think it’s pretty likely to be true.

Concept 4 — Asymmetric 7mm

Here are the results for Concept 4. Again it shows that Fusion predicts a stiffer flexure than what is actually measured. This time, Fusion predicted a 48% stiffer flexure (based on the trendlines in the plot). But the nonlinearity I discuss a bit more under the plot is ‘pulling down’ the trendline for the actual measurements. For the lower load values, the difference is in the low 50% range.

One thing I noticed in the above plot is that it appears to me that this one is entering a more nonlinear behavior at the higher load values. Notice the Fusion predictions continue to predict a linear response. If I had the correct material definition, I could presumably improve this with the Fusion nonlinear solver. But, more importantly, this also could be important for end applications. For example, if I wanted to use these diaphragm for a load cell, this nonlinearity would make calibration a bit trickier and would introduce more uncertainty into the measurements. That’s not to say it couldn’t be used for those applications, but I’d probably want to tune it to make sure it would stay within the linear range for whatever the load range is of the force sensor. However, major caveat to all of this, the above data is pretty coarse. It’s possible that the nonlinearity I’m seeing is a result of noise in the measurements. For now I’m not too worried about it, but if I want to really characterize this potential nonlinearity, I am going to need finer resolution on my test loads.

Concept 2 — Symmetric 7mm

And here’s a look at the results for Concept 2. I am pretty surprised to see that this one has the highest deviation between Fusion prediction and measured, with Fusion predicting a 92% higher stiffness. I can’t help but suspect I’ve missed something here, but I’ve double checked all the sim settings, and it all looks right to me. So I’ll keep an eye out, but barring finding a smoking gun, I just have to assume it’s within the error range I can expect.

Concept 3 — Asymmetric 5mm

This is the floppiest of the bunch. As a result, I didn’t run it to as high of a load as the others. Notice the real measurements saw a total deflection of 4mm, that’s quite a bit of material movement, so I’m pretty surprised that 1) the results seem to linear and 2) the Fusion results are only 50% more stiff than actual.

Conclusion (for now)

So interestingly, Concept 3 and Concept 4, the two with the asymmetric geometries, were 48% and 50% stiffer in their respective simulations. While quite a bit off from actual, it points to some hope for being able to apply a scaling factor to the sim results to get ballpark stiffness estimates. Would I trust this approach for a wing on an airplane I’d sit on, no, but it may very well be good enough for the goofy crap I build in the basement :)