Flexure Design

Flexure Design

I talked a couple weeks ago about how you can design fixtures and constraints for holding pieces in precise alignment. But not always do you want machine pieces to hold still – sometimes you want them to move. If you want pieces to move a long way (also called “large range of motion”), you can use things like bearings, slides, and gears. But because I work in the precision world, often I want things to move only a small amount – a millimeter or less. (This is much the same as you would use bolts or clamps for fixturing a non-precise machine, but I need to use kinematic couplings for micrometer precision).

So I thought today I would talk about designing flexures. And because I came down a little harder than I intended on theoretical engineers last week, I’ll show the modeling I use to design a flexure. (I do realize modeling and theory are useful!)

A flexure is a bearing that allows motion by bending the material (thus the name: bending, flexing, flexure). The simplest example is a diving board. When you jump on a diving board, you move in a vertical motion because the board is bending.

The motion and behavior of bending beams is well understood. The equation that governs the bending of beams is called the Euler-Bernoulli beam equation.

Euler-Bernoulli Equation

This equation will tell you the deflection of a beam under a given load. And then using the deflection, you can calculate other useful things, like the stress in the beam. (For instance, if you are too heavy, that diving board is going to break when you jump on it…)

To give you a couple examples, here is a flexure that allows linear motion (pushing on the top will move it linearly with respect to the bottom plate, as shown).

Linear Flexure (Ref:http://www.precisionballs.com/Flexures.html)

And here is a flexure that allows curved motion. Pushing on the top will move it in an arc pattern with respect to the bottom.

Curved Hinge Flexure (Ref: http://www.wa.ctw.utwente.nl/staff/d.m._brouwer)

For research recently, I had to design a flexure that would hold a square thing, and allow the square thing to move in three directions (and three directions only). Vertical z motion, and tip and tilt around the x and y axes, respectively.

Here’s the flexure I designed. The square thing I’m moving around is removable, and will go in the hole in the middle. The outer ring will be bolted securely, and then the three long curved sections are free to be flexible. Then the square thing can move in the three intended directions with respect to the grounded outer ring.

Flexure With Grounded Inner and Outer Rings

Take a look at how this moves at this link:

Flexure Deformation

I used Solidworks as my CAD program to create this part (although I’ve used ProE before as well), and then used the FEA package to analyze it (that’s the simulation .jpg picture with the mesh below).

Flexure Model Under Analysis

I wanted to make sure that under load, it wouldn’t deform more than the acceptable yield stress. Here’s a link to the simulation (dare I say, the model? :)) of the stress in the flexure under load.

Flexure Stress

The whole machine together looks like this:

Flexure Integrated Into Equipment

This week I have been in the machine shop, making all those various parts. Successful so far, although with the number of holes and bolts, I was almost surprised everything lined up correctly! The machine in reality looks like this (with the square thing on the table):

Flexure Integrated Into Equipment

So the moral of the post today is that if you need to design a moveable bearing for a small range of motion, you might try using a flexure. Happy designing!


The “flexure integrated into equipment” has piqued my curiosity. Methinks I know to which equipment is goes. 😀

Brouwer’s got class notes on designing flexures and you can also use Alex Slocum’s book or Stuart Smith’s book. Which do you prefer?

Spincoater! 🙂

Plan is to use the spincoater to cover a stamp with an ink, and then put my substrate in the flexure setup, and press it down onto the stamp. I need those flexible degrees of freedom to account for waviness or bumps in my stamp and/or substrate…

I use Slocum’s book, but I didn’t know about those other two resources. I’ll check it out, thanks!

What fabrication process did you use to get the square hole? EDM? How many of these fixtures do you intent to assemble?

I cut the whole flexure out of a flat plate of steel using a waterjet. I used a waterjet to cut out the rings with bolt holes that you see in the “Flexure With Grounded Inner and Outer Rings” as well. Since everything is 2-D, the waterjet works really well. A 13″ diameter ring would be tricky to do on a mill or lathe, and the EDM would work quite well, but is expensive.

You can look up the yield stress of the material you are using, and you definitely don’t want to be above that. But a good rule of thumb is to design for only 1/3 of the material’s yield stress.

The reason for that is that the stress concentration factor for a notch or crack is K=3, which means that if you have a crack in your material by accident, the stress there will be three times greater than the bulk stress. You still don’t want your design to fail, so acceptable stress for me is 1/3 the material’s yield point.

Did you have to design and build the flexure yourself? There’s nothing available on the market that would do the job?

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