# Methodology for Measuring the Camera’s Moment of Inertia

In the previous post on calculating damping time, I showed that in order to complete our analysis, we needed to know what the moment of inertia for the cameras in question are.  The motions of a tripod are predominantly rotational, and so instead of just using the weight of the camera and lens, we need to use the rotational analog of weight, known as moment of inertia, also known as angular mass or rotational inertia.  Rotational inertia works exactly like weight does in a straight line.  More weight = harder to speed up or slow down.  Similarly, more rotational inertia means more damping is required to stop it from oscillating on a tripod.

Calculating the moment of inertia of an object is straightforward.  Take every bit of the object, weigh it, and multiply by the distance from the center of rotation (radius) squared.  Or, in math terms, integrate the mass times radius squared.  This is great and all, but we don’t know what the mass distribution is for cameras and lenses, so we will have to measure it instead, bringing us here.

We know that tripod are near perfect harmonic oscillators, and thus adhere to the following equation regarding their oscillation frequency: $\omega = \sqrt{\kappa/I}$

where $\omega$ is the oscillation frequency, $\kappa$ is the stiffness of the tripod, and $I$ is the moment of inertia.  So if we add additional weight $I_{c}$ in the form of a camera to a known system, we would get: $\omega_{c} = \sqrt{\kappa/(I + I_{c})}$

The stiffness remains the same, so we can combine these equations to get: $I_{c} = I ((\frac{\omega{c}}{\omega})^{2} -1)$

So, if we know the initial moment of inertia of the system (which we do), we can calculate the inertia of the camera and lens simply by measuring the frequency of oscillation before and after adding the camera.

The accuracy of this measurement will be mostly dependent on lowering the amount of noise and variance in our frequency measurements.  We want to minimize the amount of initial inertia placed on the tripod such that adding the camera will result in the largest possible difference in frequencies.  However, we want to keep enough initial inertia to make the oscillation last a long time.  We therefore also want to use a tripod with as little damping as possible.  Fortunately, the perfect tripod for this recently came across my test bench in the form of the Jobu Design Killarney.

Typically I measure the frequency of a tripod by fitting a sinusoidal function to the time domain oscillation.  Coupled with an exponential decay, this is the typical method I use for measuring stiffness and damping.  For measuring the frequency only, I wanted to compare this with measurement directly in the frequency, or Fourier domain.  Here is an example of the tripod oscillation in the time domain: And now in the frequency domain: A perfect oscillator in the time domain looks like a delta function in the frequency domain.  In practice, no oscillator is perfect and there will always be some width to the frequency distribution.  Here though, we only really care about fitting the center peak.  The Lorentzian function was a slightly better fit to the data than a simple Gaussian, but they both fit to the same center frequency of 11.26 hz.  This is functionally equivalent to the 11.27 hz fitted in the time domain.  In practice I found neither method to result in particularly more accurate measurements than the other.

Now as an example, here is what happens when I place a large lens, the Pentax 67 400mm F/4, onto the tripod: The frequency of oscillation has dropped considerably, down to 10.31 hz.  Even though this is the largest, heaviest lens I have, its moment of inertia is still clearly small compared to that of the test bar, despite the lens weighing more.  This is directly a result of the deliberate weight distribution on the test bar.  The weights are placed at the ends, 33 cm away from center, while the weight of the lens is mostly at the center of rotation.  The r^2 term in the moment of inertia tends to be more important here than the actual mass.

I briefly measured the inertia for several lenses mounted on the GFX 50S, and the results with the implied damping time on the Jobu are shown below. Even with the terrible damping on the Jobu, the damping times are reasonable for the smaller lenses.  This requires a more in depth analysis, but likely means that damping is not going to be that important except for when using large telephoto lenses.  In the future we are going to want to invert this analysis and ask “how much stiffness and damping are necessary to achieve reasonable damping times?”

In the GFX with 45mm lens is a fairly normal size for a camera and the inertia is a factor of a 100 smaller than what we typically use for testing tripods.  It is going to be somewhat challenging to measure even smaller camera and lens combinations.  We may have to reduce the test inertia further.

For stiffness testing, we use enough inertial mass in the test setup that the inertial mass of the tripod itself doesn’t matter.  When using a small camera though, the inertia of the tripod should be in the same range as the camera, so we will have to take that into account when calculating damping times.  This can also result in more complicated behavior that we are modeling here, as the damping elements within the tripod are not placed to damp out the energy in the tripod itself.  Fortunately for us, when the load on the tripod is that small, the damping times are likely going to also be so small as that this more complicated behavior will be irrelevant to a practical guide on tripod use.

Stay tuned for a more comprehensive set of inertia testing.

Takeaways

• We can measure the inertia of camera and lens combinations by looking at the change in resonance frequency of a tripod after placing the camera upon it.
• For most normal sized camera and lens combinations, the inertia is quite small, and therefore damping times will be as well.

Questions

• How much inertia does the tripod itself exhibit?
• What kind of inertia can we expect for the most typical camera and lens combinations?
• What kind of damping times can we expect from typical systems?
• When is damping most important?

## 3 thoughts on “Methodology for Measuring the Camera’s Moment of Inertia”

1. Tom Mahony says:

Hi David,

I wanted to reach out to you with a tip on turning your time domain signal into a more accurate frequency
domain signal. Hopefully I’m not preaching to the choir, so if you already know all this, feel free to ignore my
email. When measuring oscillators, and using discrete fourier transforms to get the frequency content of a time
domain signal, you have to let your signal ring down all the way to 0 (or sufficiently small enough signal) so as
not to introduce apodization errors. Here are some slides that kind of illustrate the problem, though I admit the
math gets a little “theoristy” for me near the end: http://math.mit.edu/~stevenj/18.325/filter-diag.pdf (skip to the
3rd slide or so). Luckily, the author of these slides, also wrote a package you can download that fits time domain
signals and extracts the eigenmodes (frequencies) and eigenvectors (amplitudes):
The reason I think this will be especially helpful for you while testing tripods with cameras, is now your system is
no longer a single mode oscillator, it clearly has multiple oscillating modes, with different quality factors. This is
pretty clear from the last plot you show on this page: https://thecentercolumn.com/2019/08/07/methodology-for-
measuring-the-cameras-moment-of-inertia/
There is a very narrow sharp peak where the tripod’s yaw rotational mode is, and then there is some very
broadband mode that has to do with introducing the camera. It might be interesting to experiment with how this
mode gets excited (at what angle you bump the tripod or camera do you see the largest excitation of this mode).
I’m also curious to see if some camera/tripod combos end up producing coupled modes.

Anyways, hopefully my unsolicited advice can be helpful. I’m a huge fan of your site, and can’t wait to see what
you measure next.

Cheers,
Tom

1. Tom Mahony says:

I forgot to mention, there is a python module for calling this library that simplifies things greatly:
https://pypi.org/project/pharminv/

1. David Berryrieser says:

Thanks, will look into that.