Camera + Lens Moment of Inertia

In order to properly calculate damping times, we need to know the moment of inertia (MOI) for the camera and support system as a whole.  In the previous two posts, we have measured the MOI for tripods and ball heads.  The final piece is what we discuss here, the MOI for the camera and lens combination.  Under vibration, the camera rotates about the apex of the tripod.  It is not the weight of the camera that we directly need to slow down, but its rotational MOI.  Weight located far away from the center of rotation will have a much larger affect than weight located near the center.  We therefore expect longer telephoto lenses to have much larger MOI than more compact shorter lenses, even if the weight difference is negligible.

To measure the MOI, I am using the methodology described here.  Briefly, we can measure the resonance frequency of a tripod with and without the camera attached.  From the difference in frequency and knowledge of the initial MOI without the camera, we can then calculate the camera’s MOI.  I tested all of the camera and lens combinations I had available, plus a couple super telephotos that were kindly lent to me for this project.  The results are shown in the table below:

The Fuji GFX 50S is roughly the size and weight of a full frame DSLR, and the X-H1 is representative of most mirrorless cameras.  The mass of the camera though, which tends to be centered over the tripod, has very little impact on the MOI.  It is the lens and how far it sticks out from the camera that matters.  Often tripod manufacturers suggest that the tripod is capable of supporting cameras up to a specific format.  We see that this is misleading, and the format size of the sensor has very little to do with the camera + lens MOI.

For context, the MOIs of the larger tripods I tested are roughly the same as the MOI seen here for the larger normal, or smaller telephoto lenses tested here.  For the any of the larger telephotos, the MOI of the camera + lens will dominate the total MOI of the system.  Since ball heads are so compact, their MOI is negligible.

Lets take a look at the data graphically.  First up, lets just plot the MOI vs Weight of the camera + lens.

 

The red line represents a simple exponential fit, with the exponent and amplitude printed on the graph.  The weight has a rough correlation to the MOI, but we can see that its not a great fit.  A simple linear fit is even worse.  From our discussion before, we know that the length is more important than the weight, so lets look at that correlation next:

Much better.  It looks like one could make a very reasonable estimate of the MOI based solely on the length of the system.  Just from the physics, we expect an exponent of 2 from the length of the system, as MOIs are given in the form (mass * length ^2).  We see the higher exponent of 2.78 here because the mass of the camera and lens combo is also increasing as we attach longer lenses, roughly in proportion to the length, giving us an exponent that is fairly close to length^3.

The super telephoto lenses are really dominating the upper end of the graph here.  They are heavy, but not dramatically more than some of the other equipment in question.  It is the combination of their sheer size and bulk that pushes their MOI so high.

Next, lets be more intelligent about our fitting, and plot MOI vs the camera + lens weight * length^2.  Most basic MOI formulas take this form, times some constant.  While a camera and lens obviously isn’t a simple shape such as a cylinder, for our purposes, it may not be that far off.  Welcome to practical physics, where we try to get as far as possible with the simplest possible models, and cows are spherical.

 

This is really good!  The exponential fit gives an exponent of 1.067, which is within error of 1.0 (linear) for our purposes.  We would need a lot more data to clearly distinguish between the two models.

If the model of approximating the camera as a cylinder worked perfectly, we would expect the slope of the line to be about 1/12 if the rotation was about the center of the cylinder (say when using a long lens with a tripod foot) or about 1/3 when the rotation is about the end of the cylinder (using the camera’s tripod mount).

Moment of inertia rod center.svg

{\displaystyle I_{\mathrm {center} }={\frac {1}{12}}mL^{2}\,\!}

Moment of inertia rod end.svg

{\displaystyle I_{\mathrm {end} }={\frac {1}{3}}mL^{2}\,\!}

Instead we find that the slope is much closer to 1/8, which is in between the two models.  This makes sense.  When using a long lens with a tripod foot, the weight is not evenly distributed along the length, but concentrated at the ends where the camera and largest pieces of glass lie.  When using the camera’s tripod mount, a portion of the weight and length lies behind the tripod mount in the form of the viewfinder and screen of the camera.

In summary it appears that we can get a reasonable estimate of the MOI for a camera system by using the formula:

MOI = 1/8 Weight × Length²

Now that we have good data for all of the parameters in the damping equation, the next step will be to have some fun and hopefully answer questions such as:

  • What is the damping time for a given set of equipment and conditions?
  • How much stiffness and damping do I need for my (insert telephoto lens here)?
  • Which is more important, stiffness or damping?

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