Two Methods to Measure MOI

In the last post I described the trifilar pendulum apparatus for measuring moment of inertia (MOI). The controls software requires MOI as an input parameter so it can calculate the phase plane trajectories. Without knowing the MOI, the flight control computer won't know how much the jets will cause the HAPP to rotate - the jets would be "firing blind" so to speak.

I also explained that once we know the MOI, we can solve backwards and find the jet force produced by the 3D-printed nozzles. This may be more accurate than the jet force measurements we did with the force-sensitive resistor pads on the static test fire stand. An accurate value for jet force is also required by the control system.

So how'd it go?

Using the Arduino-based IMU sensor package and software sketch I developed, it was easy to identify the period of oscillation for the prototype HAPP hardware loaded onto the pendulum platform: 2016 milliseconds. Using this period, and plugging in the physical dimensions of the test setup, I got a moment of inertia around the z-axis (J_zz) of 0.1815 kg-m^2. Stepping carefully through the various measurements I calculated a margin of error of +/- 0.9%. That's pretty tight, and it validates my guess that we could get error percentages in the low single digits.

Personally, I don't have a good "feel" for moment of inertia. Answer quickly: Within a factor of 10 (or even 100), what's the J_zz of a typical passenger car? Couldn't tell you unless I looked it up. Similarly, it's hard for me to see the number 0.1815 kg-m^2 and judge if it's even approximately correct. Can we validate it with another method? It doesn't need to be super-accurate - within 20% or 30% would be fine - just close enough that we know the trifilar pendulum method is more or less correct.

Meet Mr. Rubber Band. From Hooke's Law we know that the pulling force exerted by this band will be more or less proportional to how far it's been stretched. I used one of the trifilar pendulum's hanging scales and measured pull force for different amounts of stretch to obtain the following relationship. The black line is data and the red line is a linear fit - pretty tidy for some rough data.

rubber band force.jpg

Now we have a known force that we can apply tangentially to the HAPP arm as shown in the picture below. By measuring the resulting acceleration with our on-board IMU instrumentation, and then applying our old friend the Euler equation for rotations, we can back-solve for J_zz, the moment of inertia.

The HAPP project utilizes only the most sophisticated technologies: Cardboard box, millimeter scale, and an old rubber band. Nozzle v6 points left and v3 points right.

I expected this method to be fairly sloppy, but after several trials, the average J_zz measured by Mr. Rubber Band's test was 0.1839 kg-m^2. That's only a 1.4% difference versus the nominal value from the elaborate trifilar pendulum method! Conclusions:

  1. The pendulum method is good.
  2. The pendulum method was a lot of work that wasn't really necessary!
  3. I never would have believed the rubber band results if that's all I had, so perhaps the pendulum wasn't a total waste of time :-)

Of course we'll have to come back and measure J_zz again once the HAPP has been fully constructed. We'll also need to measure moments of inertia along the other two principle axes, as well as the three cross-axis products of inertia, thereby filling out the inertia tensor. But at least we've validated the trifilar pendulum method pretty thoroughly.

Next post: Characterizing the relationship between jet force and manifold pressure.