# Tripod Moment of Inertia

In the previous post, I demonstrated my methodology for testing the moment of inertia (MOI) for a tripod.  Initially, we were trying to make the assumption that the inertial mass of the tripod was irrelevant towards calculating the damping time of a vibration.  The data in this post suggests that this is not the case, and the inertial mass of the tripod will likely be larger than that of the camera when using all but the largest telephoto lenses.  Thus, for calculating damping times, it will be critical to add the mass of the tripod, head, and camera all together.

Moment of inertia is the angular equivalent of weight, or mass.  The movements of a tripod are inherently rotational, so it is important that we analyze the damping problem in the correct way.  Similarly to how it is more difficult to slow down a heavier object, it is more difficult to damp out a rotational vibration in a system in that has a higher MOI.  Unlike mass, MOI is different based on the axis that the object is rotating.  Here, I am only measuring yaw MOI as it is the only axis about which I can have confidence in my measurements.  Also, because the camera is typically placed high above the center of rotation, the dynamics are very different and the weight of the camera will be much more important.  More on this later.  For now, we are only talking about yaw vibrations.

As it takes me roughly 20 minutes to test the MOI of a single tripod, I was only able to test a small number, which I chose to be a somewhat representative sample across different weights.  The results are below:

Its no great surprise that heavier tripods have higher moments.  The ratio column is simply the Inertia/Weight.  We can immediately see that there isn’t a direct linear relationship between weight and Inertia.  This is simply because heavier tripods tend to also be taller, and the taller the tripod is, the further the legs splay out, resulting in more MOI.  We can also see this as a result of leg angle.  The TFC-14 has a greater MOI despite weighing less than the very similarly constructed LS-284C.  The narrow leg angle on the 284C reduces its MOI (but also its stiffness).  When looking at the ratio, there are two significant outliers, the 3 Legged Thing Leo and the Feisol CT3472.  These tripods are quite heavy for their height, and very light for the height, respectively.  Again, a short tripod’s legs don’t extend out as far from the center of rotation, and for MOI, radius from the center tends to matter more than weight.

For context, I measured the MOI of the Fuji GFX 50S and 45mm lens at about 0.004 Kg*m^2.  So the MOI of the tripod will almost always exceed that of most normal sized camera and lens combinations.  This will invalidate our damping time calculations, as those assume that most of the MOI comes from the camera and ball head placed atop the tripod.  If the damping is occurring within the legs, then our calculations should still work.  However, many tripods get a significant portion of their damping from the rubber pad on the top plate.  This rubber pad will do nothing to damp the energy contained in the legs themselves.  Its efficacy will be grow as the MOI on the tripod increases.

Because the tripod MOI will be so important for damping, we will want some way to estimate the tripod MOI from things easier to measure, such as the height and weight.  To do this I am going to fit a couple quantities to the formula:

$MOI = A*(tripodmetric)^{exponent}$

First, lets use weight as our tripod metric, and we get:

Here, I have also plotted a straight up linear fit (exponent = 1) and we can see that it doesn’t describe the data well at all.  The exponential fit is reasonable, but we can still see the significant outliers of the 3LT Leo below the red exponential fit line and the Feisol CT3472 above the line.  An exponent around two is consistent with what we expect.  If the tripods were identical aside from their weight, we would expect the exponent to be straight up linear.  But as we said before, weight correlates with height, and height has a much stronger relationship to MOI.  So next, here is MOI vs Height:

We see a much larger exponent, but now we aren’t taking into account weight.  Also, the range in heights is pretty low compared to the range in weights, making this fit much more questionable.  This plot is less useful.

In theory, we could try a fit placing an exponent on both height and weight, but frankly, we don’t have enough data to get meaningful results out of such an approach.  Instead (after some experimentation) lets fit the exponential function to MOI as a function of the tripod’s weight*height^3.

Not too bad!  I have thrown the linear fit back in here to show that it is functionally the same.  This demonstrates that we are likely using the correct exponents on our height and weight factors.  So we have reduced this to a single parameter fit, which is the slope of the line.  So, moving forward, we will use the following formula for tripod MOI:

$MOI = 0.00429 weight \times height^{3} kgm^{2}$

where the height and weight are given in meters and kilograms.  I’m sure we could get more accuracy by bringing leg angle into this, but that won’t be necessary.  This is plenty good enough for our purposes.

Takeaways:

• Tripod MOI is larger than expected, and will be important to incorporate into our damping calculations
• Tripod MOI can reasonably be calculated as a function of height and weight.

Next up: