3D Printed Springs - Part II

Variation in Layer Spacing

In Part 1 of this series, I looked at how the spring rate of the “ribbon” design varies with the width of the spring. There are a few small corrections.

For the time being the plan is to next investigate the variation as a function of the thickness of the spring. Hopefully what we can see after running the final set of parameter tests that the parameter variation is decoupled and that each value can be changed independently.

Update to width variation

I noticed that in the previous post I correctly labeled my chart as “spring rate” or stiffness. The plot is actually displacement as a function of ballast weight - which is really a representation of spring compliance.

The spring stiffness appears to be a linear function of the overall width

The most interesting observation is that when looking at stiffness, instead of compliance, the stiffness of the spring does in fact linearly as a function of width, confirming the hypothesis that the stiffness can at least partially be accounted for by modeling the spring as a cantilevered beam.

As a refresher, using some basic statics equations to calculate the bending stiffness of a cantilevered beam we’d expect to see

k = Ebh^3 / (4l^3)

Assuming that, E, h, and l are constant, we would hypothesize that the stiffness k is a linear function as we’re only varying b. This in fact now appears to be the case as the linear regression provides the best fit over the parameter sweep. Neat!

Spacing Results

In this experiment, effectively only the radius of the half-circle is being varied. If we consider the half-circle portion in the same way we’d consider a tube, then the stiffness should somehow be a function of the moment of inertial of the cross-section. For a tube, we could use

I = (π*h/2) x (d_outer^4-d_inner^4)

While the above isn’t strictly correct since we only have the half-circle c-shape, it’s at least a somewhat reasonable starting point to assume that the stiffness will be a polynomial function.

A cubic fit!

Conclusions

This experimental series is turning out to be pretty exciting so far. It was great to discover the error that lead to the discrepancy between the analytical hypothesis and the experiment results regarding the width of the beam effects. Now adding in the effects of the layer spacing the empirical model is coming together. Granted, this will only be valid for the relatively small parameter sweeps.

Current model for the ribbon spring stiffness

k = (819 - 84.4s + 4.01s^2 - 0.0808s^3) + (11w + 27.1)

Next up will be to look at the variation in the thickness of the spring and how it plays into things.