A major driver of this increase in stabilization performance has been in the gyroscopic sensors used to measure the vibration and control electronics used to correct it. For these types of applications, the accuracy of the zero offset of the gyro sensor is often the limiting factor. Even when perfectly still, noise will cause the sensor to read a small but non-zero value, which will in turn cause the stabilization system to drift. The smaller the drift, the longer the stabilization system can hold an image steady. If there were no noise on the gyro sensor and the stabilization system worked perfectly, it should in theory be able to provide an infinite number of stops of benefit. However, at somewhere around 6 stops of shutter speed increase, the limiting factor stops becoming the electronics, and instead becomes the rotational motion of the Earth!

To illustrate , lets imagine that you are somewhere on the Earth’s surface, pointing the camera due East or West. For simplicity, lets assume you are on the equator, but your latitude doesn’t actually matter for this analysis. There are 86,400 seconds in a day, so Earth rotates at a rate of 2π/86400 radians/second, or 7.27*10^-5 rad/s. That means that your subject, which is presumably stationary on Earth’s surface as well, is rotating at this rate. Your camera, which is using its IBIS system to attempt to keep everything as still as possible, may not realize that you are rotating with your subject and will instead try to zero out *any *rotation of the camera, including that of the Earth. More technically, the camera is trying to maintain stability with respect to an inertial reference frame, which by virtue of the Earth’s rotation, you and your subject are not.

The camera will of course move with the Earth and your subject, but will rotate relative to the subject. If the camera is pointed east as shown in the figure above, it will rotate up relative to the subject. One may now wonder why the IS system can’t just cancel that out.

Lets say that you have an exposure time given by the variable T. IBIS will therefore cause the image to move by a distance on the sensor given by:

Using our typical standard for the maximum allowable blur to be 1 pixel width, we get the maximum allowable shutter time to be

A 24 megapixel full frame camera will have pixels that are 0.0059mm wide. Plugging this into the above equation, we get

If we take 1/Focal Length as the standard advice for the maximum shutter speed you can use to achieve sharp images, this equation says we can use shutter speeds 81 times as long with stabilization before Earth’s rotation blurs the image by one pixel. In more familiar terms:

Stops

Neat. Of course not all cameras have the same 0.0059mm pixel pitch, but the rule for the maximum hand-held shutter speed will vary accordingly, so this 6.3 stops will be consistent across most cameras. We can turn this around and say that in order to provide 6.3 stops of stabilization, the system must be able to measure and compensate for rotations of the camera as slow as the Earth’s, or 360 degrees per day. That’s pretty impressive.

Presumably, a calculation like the one above or very similar was the inspiration for the claim by Olympus Camera:

“6.5 stops is actually a theoretical limitation at the moment due to rotation of the earth interfering with gyro sensors”

Since then the plot has thickened. Olympus has released the OM-D E-M1X which claims to have 7.5 stops of stabilization, and Panasonic claims their new 70-200 f/2.8 has 7 stops of stabilization. How could this be? Without access to the data used to perform the CIPA stabilization rating test for these pieces of equipment, it’s difficult to say for sure. In theory though, there are some possible ways to work around the limitations placed be Earth’s rotation:

- Use the camera’s GPS, accelerometer, and compass to calculate exactly where it is pointed and its latitude. With this information you could calculate the necessary offset and program your stabilization system to compensate accordingly.
- Use a high pass filter on your stabilization system. The rotation of the Earth is very constant, so if your gyro sensor should measure a small but constant rotation. In comparison, your hands shake the camera all over the place, so an engineer could tell the sensor to only compensate for this later kind of motion.

The first isn’t a good solution for many reasons. Don’t have GPS signal? Shooting next to a magnet? Your system won’t work. The second solution is much more plausible, but still very difficult. The user would have to be pointing the camera at the subject for long enough such that the drift in their aim at the subject is smaller than the drift from the rotation of the earth. This is also implausible. What is concerning though, is that this second method is one that could work very well to cancel out Earth’s rotation on the CIPA specified stabilization test apparatus.

For the CIPA stabilization test, the camera is mounted to a vibration platform. The camera is then moved by the platform in a way that is determined by a CIPA provided waveform designed to mimic the motions from hand-holding a camera. The setup is shown below:

In this test setup, the camera is mounted to a very stable platform and held there for a long time before and after the test images are being taken. In theory, this could make it much easier to for the camera to identify and filter out any effects from Earth’s rotation. If the waveform has no very low frequency components, it would be easy to identify any low frequency motion as Earth’s rotation. Without knowing exactly what waveform is given to camera manufacturers to perform this test though, it’s impossible to draw any meaningful conclusions. It would be great if we could get some independent verification of the stated stabilization values, but CIPA seems to make that deliberately difficult. As quoted on their application guide:

- The applicant must be a corporate organization.
- The applicant must understand that the standard and measurement kit cannot be provided to individuals, universities, or any other parties for research purposes.

This seems unnecessarily restrictive to me and makes independent verification and analysis of the test impossible.

This has been a long way of saying that I don’t know if Olympus and Panasonic have found a way to compensate for the Earth’s rotation in their stabilization systems, breaking through the 6.3 stop limit, or if they are achieving 7+ stops through oddities of the CIPA rating test. If anyone has one of these cameras with 7+ stops of stabilization and some time on their hands, it would be really cool to show the camera breaking through the Earth rotation barrier. If the cameras can’t, and you could show that Earth’s rotation is actually the limiting factor on the stabilization system, well that would be pretty cool too.

]]>Tripods are relatively simple and their performance can be accurately summed up in their stiffness and damping characteristics. Tripod heads however are more complex and varied in their design. It is thus more difficult to capture the performance of the head in a single metric. The maximum torque that can be applied to a head before it slips, the shift in framing when locking the head, and the geometry of construction all contribute to the performance in a way that isn’t captured by looking solely at the stiffness of the head. With that disclaimer out of the way, stiffness is still the most important property of head when it comes to supporting weight. Stiffness is what keeps things still, the difficulty of which increases with the size and weight of the gear used.

If we plot the manufacturer’s weight rating vs the stiffness of the head measured on this site, we get the following array:

What we see is similar to the situation for tripods. Any given weight rating represents a huge range of possible stiffness’. While the data here is of course not actually random, I could find no pattern to predict the stiffness performance of a head based on other factors such as the cost or reputation of the manufacturer. The ball diameter and more importantly the width and height of the stem do correlate strongly to stiffness, but that is the subject for another post. All we are trying to do here is discredit the weight rating. If the measured stiffness data isn’t available, like with tripods, it is much better to focus on the geometry of the construction rather than the manufacturer’s weight rating when trying to evaluate performance.

To highlight the absurdity of the situation, compare the performance and weight rating of the 3 Legged Thing (3LT) Airhead Switch and Feisol CB-70D. The Feisol is the largest diameter ball head I have tested and by far the stiffest. It is an absolute beast that is only at home on the largest series 4 & 5 tripods. The Airhead Switch is a compact and lightweight ball head designed for travel tripods. It isn’t a bad head and sports a reasonable stiffness/weight ratio, but is nearly an order of magnitude less stiff than the Feisol.

In addition to stiffness, there are a couple of other factors we have referred to previously that could clearly be related to the effective load capacity of a head. Before addressing those, lets dismiss the one that clearly has no direct correlation. With tripods, enough weight can plausibly splay the legs out so much that they break. While this is not the preferred way to generate a weight rating for tripod legs, this is the strategy some companies employ. For heads though, it is implausible to think that we could physically destroy one simply by loading weight on top. Of course in possible with enough pressure, but metals such as aluminum have incredibly high compressive strength as can be seen here. I’d love to see what happens to a ball head under a hydraulic press, but it wouldn’t further our practical understanding of camera support.

If the force isn’t directed straight down on the head, we would expect the first failure point to be the friction device locking the ball or tilt axis in place. When the camera is positioned off to the side of the head, it will generate a torque about the rotation point (usually the center of the ball). A heavy enough camera could generate enough torque to break the friction holding the ball, causing it to flop to the side. I have contacted several manufacturers who use some variation on this method. As shown in the image below, Acratech has publicly shown this as their test setup.

Technically, this is a torque specification, and to translate to a weight capacity specification they assume a certain distance that the camera will sit from the ball when attached. This becomes problematic as larger and heavier optics tend to have big lens feet that push the weight of the imaging system further from the ball than you find for lighter setups. Overall though, this is at least a rational basis on which to assign a weight rating.

Unfortunately, it still produces some irrational results. Take for example the Highline Ball Head by the Colorado Tripod Company, which is rated by this method to hold 70 lbs. The Highline is a reasonable head, but no sane photographer would ever attempt to use it to support 70 lbs. of gear. Even with an infinitely stiff tripod, the vibrations allowed by the head while supporting 70 lbs would take many minutes to damp out.* This is unacceptable for any kind of workflow. I am only picking on the Highline head here because to their credit, the Colorado Tripod Company has talked about how they rate their heads. In practice, no ball head is appropriate for holding significant amounts of weight simply due to their ergonomics.

Ball heads place the camera above the point of rotation, which is naturally located at the center of the ball. This creates an unstable system in which the camera is prone to flop over to the side. The effect is mitigated by having the camera placed as close to the center of rotation as possible and by increasing the diameter of the ball. The closer the camera is to the center of rotation, the less torque will be generated from gravity when the camera moves off to the side. It will then be that much easier to correct for small movements and handle the weight of the system. Remember, when positioning the camera with a ball head, *you* are part of the the camera support system. Acratech has recognized this limitation and advises customers on their site:

“Any of our heads will easily hold the weight of the big telephoto lens but it is awkward with ANY ball head when re positioning something large and heavy on a round object. Think of an elephant balancing on a ball.”

Correctly adjusting the tension on the ball head can help. Here is where the size of the ball matters. A larger ball has more surface area to provide the correct amount of friction without locking up.

Geared and pan-tilt heads are ergonomically more adept at supporting weight than their stiffer ball head counterparts. With a ball head, the yaw, pitch, and roll axes of adjustment all lock down simultaneously. This can be convenient, but with a larger and more cumbersome load on the head it can also be frustrating. A pan-tilt head allows you to adjust each axis independently without disturbing the others. A geared head does one better and allows for minute and precise adjustments as well. Better still, the design of many geared heads such as the venerable Arca Cube place the center of rotation at the camera, reducing the effective inertia mass of the system and taking the weight out of the users hands in a similar way to a gimbal. Most people find these tools much more capable of efficiently positioning larger loads than ball heads. In this case, the stiffness number alone does not adequately capture the overall performance of the head. Stiffness is of course still important, and the only thing the holding the camera still against external forces.

A quick note on gimbals: The heaviest photographic gear is typically used on a gimbal setup. The point of these is to ease the handling of a long lens by placing the center of mass at or below the rotation point of the head. Absolute stiffness isn’t as important because gimbals are usually used to support gear, not lock it in place. The weight rating may here is worth paying attention to, much more so than for other types of heads. But again, I am unaware of any universal standard by which the weight rating is created. Take it with a proverbial grain of salt.

The summary for this article was dropped in my lap after Manfrotto finally got back to me on their weight rating system:

“Unfortunately, there is no industry standard for determining the weight capacity of tripods and heads. Each brand goes through their own process to determine the weight load of their products. With that said, the actual method is proprietary and I am not permitted to go into detail about it.”

When I see the term proprietary in this context, I can only assume a manufacturer means “We don’t actually have a rigorous method for testing and don’t want to admit we just make it up.” Given that the industry doesn’t have any standard for measuring weight ratings, please don’t refer to the ratings as if they had any useful relation to the support of photographic equipment.

*We are even making some optimistic assumptions. To make this calculation yourself, use the damping time equation from “Calculating Damping Time“, and an estimation of MOI for a 70lb piece of gear from here. Fill in the rest with some reasonable assumptions.

]]>For example, the Sirui T-025X is one of the smallest and lightest travel tripods available, and is well regarded for those qualities. The stiffness unfortunately suffers as a result, making it one of the least stiff tripods I have tested. Yet, it is rated to hold 6kg, or 13.2lbs, roughly the weight of a 600mm f/4 lens and attached pro DSLR. While the T-025X will physically support such gear without collapsing, the tripod was clearly not designed for such use.

I am making the assumption here that the weight rating applies to photographic and video equipment, as this is the intended use. Heavy imaging equipment also tends to be physically large and bulky. As the vibrations on a tripod are predominantly rotational in nature, the appropriate measure of load is the rotational moment of inertia (MOI) of the camera and lens, which depends on the size of the equipment as well as the weight. A tripod may be able to hold a 10 lbs of exercise weights easily, but a 10 lb super telephoto lens will feel incredibly unstable. So, even if there were a standard way to calculate weight ratings, it is a poor metric to use.

I have argued on this site that the tripod’s stiffness is the single best metric for judging its stability¹. Stiffness is by definition the ability of the tripod to perform its central function, keeping the camera still. Stiffness can be measured accurately, and is directly comparable across all tripods. Armed with the data on this site, we can see just how meaningless the the weight rating is as a metric for stability. The following plot shows the weight rating versus of stiffness for every tripod I have tested²:

It is not completely random, there is some correlation, but for the purposes of choosing a tripod, it may as well be random. For tripods of a given weight rating, the stiffness varies by roughly a factor of 5. Tripods lower and to the right of the graph are rated more conservatively. Those in the upper left have been rated ~~by the marketing department~~ generously.

If you are a tripod company trying to figure out what your weight rating should be, 10 kg for every 1000 Nm/rad of mean stiffness seems to be on the conservative side of average for how tripods are currently rated. 20kg / 1000Nm would be on the more optimistic side. If I had to give a weight rating for tripods, I would say something like 3 kg (~6.6 lbs) for every 1000 Nm/rad of mean stiffness is appropriate for general use. Before someone comments “But I put X heavy gear on my lightweight tripod and it held it just fine!” or “I really struggled with my 100-400 on my large tripod in the wind.”, yes, I agree with you, weight ratings are dumb, and produce dumb results. 3 kg/1000Nm is just the metric that seems to best approximate my experience with tripods and photographic equipment.

Even within a single brand, there isn’t total consistency. The following chart breaks out some specific tripod companies by color.

The premium brands tend to be more conservative with their weight ratings, but there is still significant variance. For example RRS gives a weight rating to a whole series of tripods. All of their Series 2 tripods are rated for 40 lbs (~18 kg) but there are significant differences in stiffness between the models.

The most egregious offender in terms of exaggerating the tripod’s weight capacity is very clearly the 3 Legged Thing Leo. It is one of the weakest tripods I have tested, yet has the 4th largest weight rating at 66 lbs (30 kg). I reached out to 3 Legged Thing for comment on their anomalous weight rating, and this is their reply:

Leo is tested with weight loads until he breaks, and can handle much more than 30Kg although we’d not advise it, so 30Kg is the safe weight limitation.

Moving past the rather morose consequence of personifying a tripod when testing it to destruction, this is a defensible definition of a weight rating. However, clearly not one that has any reasonable relation to supporting photographic equipment. The largest commercially available camera lens is the Sigma 200-500 f/2.8, which weights in at just over half the weight rating at 15 kg. If you mounted this on a Leo tripod, it would flop about like a fish out of water, and you wouldn’t take your hands off of it for fear of a slight breeze toppling it all over. But the tripod legs wouldn’t break, even if you mounted a second 200-500 right next to the first.

Really Right Stuff takes a more realistic approach in their tripod guide. They de-emphasize the physical weight rating of the tripod (while simultaneously showing the owner of the company hanging from their smallest tripod) in favor of a rough focal length limit.

A true support will not only be rigid, free-standing and load-bearing, but should also dampen enough vibration to allow the camera and lens to resolve fine detail at their maximum clarity and sharpness. …

Most photographers, however, are using gear that weighs much less than those [weight] ratings. Technically the smallest tripod, our TFC-14, could hold a 500 or 600mm lens without collapsing. So why don’t we recommend that combination? It all comes back to vibration …

This closely mirrors my experience in testing for this blog. Focal length and overall bulk have a greater correlation to the demands placed on the tripod than does weight.

Peak Design has also publicly acknowledged the problem with weight ratings during the release of its travel tripod

[About] our 20lb weight rating. Here’s what that means: you can put a 20 lb. object on top of the Travel Tripod and it will still safely function. Can it hold more without collapsing? Yeah, a lot more. It takes about 85 lbs. of downward force before the center column starts to slip. Do we recommend using the Travel Tripod as your go-to solution for holding 20 lb. objects? Not really. Just because it can hold 20 lbs. doesn’t mean it’s optimized for it.

… An 8 lb. rig with a short lens may be more stable than a 5 lb. rig with a long lens. …

This is also a sensible interpretation of what rating should mean. I have not seen such discussion on the actual meaning of their weight rating from other tripod companies. If you have seen more information the issue that I have missed, please let me know.

If weight rating is a poor metric for stability, guidance, and comparison, we are going to need something to replace it. The weight rating has been so pervasive in the industry simply because it is easy to understand, and for the consumer to measure the weight of their equipment. While an MOI rating would be more accurate, it isn’t practical for the consumer to measure the MOI of their equipment (though it can be estimated accurately). It also doesn’t account for the magnifying effect of long focal length lenses and high resolution sensors, which require more stability. A maximum focal length recommendation could also work, but has its own issues. A 400mm f/2.8 clearly requires a very different support than a much lighter and more compact 400mm f/5.6.

This blog does not yet have a gear limit recommendation for a given tripod. It is important to me that, unlike the current weight rating standard, any such recommendation is backed by data and relates to the real-world experience of using a tripod. While we have some preliminary data that moves us towards that goal, we simply aren’t at the point yet of making a specific recommendation that photographers will rely on.

In the meantime, stiffness and damping can be measured with accuracy, are the best metrics for characterizing tripod performance, and are useful for comparing tripods to each other. You can leverage your experience with one of the tripods tested here (or similar) to get a reasonable feeling for how the others will perform based on the difference in their stiffness. Photographers expect very different things from a tripod based on their gear and the conditions they shoot in. Arctic winds necessitate a very different support than the controlled environment of a studio. A small travel tripod can be infinitely better than the sturdy beast that gets left at home. A $5 cable release can make a bigger difference than a $1000 tripod. An accurate accounting of your requirements for a camera support and a little common sense will get you a lot farther towards choosing a tripod than a meaningless weight rating.

**Takeaways**

- Do not use weight rating as a metric for comparing tripods.
- Do not use weight rating as a metric for how much photographic gear you can comfortably use on a tripod.
- An accurate gear recommendation for tripods is complicated and depends on the gear MOI, focal length, conditions, and technique. We don’t yet have the data to precisely make this assessment. Leverage your own experience and needs rather than a weight rating when choosing a tripod.

**Future Work**

- Continue collecting data that leads us to a reasonable recommended gear limit for each tripod tested here.
- Decide what approximations need to be made to assure that the recommendation is simple yet reliable.

Footnotes

- Damping also plays a very important role in tripod performance. The relative value of stiffness vs. damping under specific circumstances is currently a subject of investigation. We can say though, that damping is primarily of value when using larger, long telephoto lenses, while stiffness is always important.
- Note, the stiffness measured here is at the full height of the tripod. Often we height adjust the stiffness for comparisons like these, but in this case, if a tripod is rated for a particular weight, it should be able to carry that weight over the entire height range.

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).

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?

Below are the raw results for two of the largest ballheads I have at my disposal, the RRS BH-55 and the Feisol CB-70D. I don’t normally post the raw results, since, they aren’t that interesting, but in this case they are critical to showing why this test hasn’t worked well.

The leftmost column shows ten trials of measuring the resonance frequency of the tripod and test rig without the head. The other two columns show five trials for each of the two heads I tested. Note how little change there is. The standard deviations of each data set would overlap with the means of the other data sets. In theory we could just build up enough statistics to get a reasonable standard error on the measurement, but in practice this incredibly time consuming and highly dependent on the assumption of Gaussian statistics.

Compared to the camera and tripod MOIs that we have measured so far, the MOI of the heads measured here are insignificant. Despite being heavy, the heads are compact with their mass located close to the axis of rotation. For example, lets look at the CB-70D and calculate its MOI by approximating it as a solid cylinder. The MOI for such an object is given by:

From the CB-70D review page, we can see that the mass is 0.971 kg and the radius as measured from the base diameter is 4.2 cm. That gives us an MOI of 0.00086 kgm^2. This is in the ballpark of what we measured above. Given the error in our measurement and the crudeness of this approximation, these results are in enough of agreement to form a reasonable sanity check. Because I am barely able to measure the MOI for the largest heads I have, any of the smaller ones are going to read effectively zero. Going forward, if we need to use the MOI of a head for any reason, we will simply make a rough approximate calculation.

I also attempted to measure the pitch MOI, with similarly poor results. But lets talk about pitch MOI briefly. For yaw, the axis of rotation extends up through the middle of the tripod and so the head will always be centered on that axis. For pitch, the axis of rotation is horizontal and extends roughly through the point in space where the imaginary lines representing the center of each leg intersect. Typically this is somewhere in the middle of the head, depending on the width of the top plate and apex. For tripods with a center column though, the MOI of the head and camera in the pitch direction will increase dramatically as the center column raises them up and away from the center of rotation. There is much more to investigate on this, and will be a theme in our future discussions on center columns.

**Takeaways**

- Ballhead MOI in the yaw direction is basically negligible.
- Ballhead MOI in the pitch direction depends mostly on how far above the center of rotation the center column takes it.

Next up: Camera and Lens MOI

]]>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:

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:

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:**

- Head MOI
- Camera / Lens MOI
- Damping times

For my stiffness tests, I use a large enough angular mass to do the test such that the MOI of the tripod is negligible. In real world conditions though, the MOI of the tripod is important and may very well be larger than that of the camera. To measure the tripod’s MOI, I used the typical equation for the resonance frequency of a system and broke out the MOI term into and .

Now by varying and observing the resulting change in we can fit the data to the function above and extract the MOI of the tripod. This method is highly dependent on having a very accurate estimate of the MOI of the test mass, which in our case here is questionable. Using simple MOI formulas for basic objects, I suspect that we can get within 1% or so. Again, this is plenty accurate for our stiffness tests as it would throw off the test by that percentage. Here though, the error would be added to our result, which can throw it off wildly.

To test this method, I picked out one of the smallest, lightest tripods I have at my disposal, the Leofoto LS-224C. If any tripod should display a MOI close to zero, it is this one. I used a much lighter MOI for the test mass than I typically do to maximize the effect that the tripod’s MOI should have on the data.

The raw results are shown in the table to the right. The test MOI is shown in the leftmost column, then the resulting measured frequency. The implied stiffness is the yaw stiffness of the tripod calculated from the MOI and frequency of that row. I typically use an MOI of about 0.4 kg*m^2 for tripod stiffness testing, and at that mass, we measured a yaw stiffness for the LS-224C of 230 Nm/rad. As we reduce the MOI of the test mass, we see the implied stiffness drop indicating that the MOI of the tripod is becoming relevant.

To actually calculate the tripod MOI, I fitted the data to the resonant frequency equation and allowed python’s curve fitting function to calculate the tripod stiffness and the tripod’s MOI. The results are displayed below:

First off, the fitted tripod yaw stiffness is at 230 Nm/rad, exactly what we had calculated previously with the larger MOI test mass. This is a great sanity check on our methods. Second, the tripod MOI is fairly small at roughly 0.003 kg*m^2. This is a fairly plausible value. It is small but positive, exactly what we expect to see for this tripod. It means that we are successfully calculating the test moment to less than this value in error. We should not take the 0.003 kg*m^2 value for the tripod as particularly accurate. It is probably within about 50%. What it does mean though, is that we can take our future measurements for the MOI of larger tripods to be reasonably accurate.

For reference, in the previous post I measured the GFX 50S with the smallest lens at 0.004 kg*m^2, so in this case the tripod MOI is less than that. Given how small the LS-224C is, it is reasonable to expect most tripods to have significantly higher MOI, and thus dominate the total MOI of the system (camera + head + legs). For the purposes of damping calculations, it will thus be imperative to get reasonable measurements of tripod MOI. It remains to be seen whether or not I need to calculate the MOI for each tripod during testing. I certainly hope not as this is a rather time consuming process.

**Takeaways**

- Fitting the test MOI vs frequency data yields reasonable results, and will result in reasonably accurate MOI measurements for larger tripods.
- The tripod’s MOI is large enough to be a significant factor in calculating damping times.

**Questions**

- So what do the MOI’s for other tripods look like?
- What about MOI in the pitch direction?

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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:

where is the oscillation frequency, is the stiffness of the tripod, and is the moment of inertia. So if we add additional weight in the form of a camera to a known system, we would get:

The stiffness remains the same, so we can combine these equations to get:

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?

Our focus on stiffness in understand the effects of wind is well founded. Wind varies on a much longer time scale than that of the resonance frequency of the camera-tripod system. This means that damping will do nothing to help keep the camera still and the image sharp during longer exposure times. It will of course still help the camera return to a neutral position after a push from the wind more quickly. This is helpful and can reduce the total amount of motion blur in the image, but cannot eliminate it in the way that stiffness can. This dynamic motion is interesting and will be the subject of future research. For now though, I want to start simply and develop a framework for much more basic problems in damping.

Many informal tripod tests floating around the internet revolve around the settling time for a tripod after an impulse is applied. The test will usually be conducted as such: Place a large lens on the tripod and then give it a little push. Then measure the time before the vibration in the viewfinder settles. This method does correlate with tripod performance, but not in a way that clearly distinguishes between the effects of stiffness and damping or the value of each. It is neither repeatable nor well defined in what it means for the image in the viewfinder to be ‘still’. Here, we are going to formalize this style test and relate it back to real world tripod use.

When handling the controls on a tripod mounted camera, we exert more torque onto the camera than it will typically receive from wind. We cannot reasonably expect the tripod to hold the camera still to within the pixel level under such forces. This is mostly true for telephoto lenses. We all know it is possible to obtain sharp images with a physical shutter press for shorter focal lengths and shutter times. We have also all had to wait for the camera to settle after handling the controls with a longer lens. It is this settling time that we want to ensure is not a hindrance towards workflow. Exactly how much settling time is acceptable will of course vary between photographers, but we will need to accept some standard amount of time. I will propose 1 second, as cameras typically have a 2 second timer and we need to leave some small amount of buffer time for the finger to come off of the shutter after pressing it. If a different amount of time seems more reasonable to you, please say so in the comments. I will adjust the standard to consensus.

Drawing from the test methodology post, after an momentary impulse of torque such as the touching of the camera, the angular position of the camera will be given by:

where

and is the initial amplitude of the vibration.

Because we are only focused on the decay of the vibration, we will throw out the cosine oscillation term and are left with:

We are going to assume that the initial displacement (or initial amplitude of vibration) is caused by a torque that is slow compared to the resonance frequency. A good example of this would be touching or nudging the camera. A bad example would be the mechanical shutter, which moves very quickly. So, we get:

where is the torque from the impulse and is the stiffness of the tripod. Following our previous standard for pixel level sharpness, we want the amplitude of the vibration to reduce below:

where is the focal length of the lens and is the pixel width. Putting everything together we get:

where is the damping constant and is the moment of inertia. Solving for we find that the time required for the tripod to damp vibrations down below the pixel level is:

This is far less neat than our calculations of required stiffness. We find that the amount of damping time depends on a lot of factors. Even if you ignore the pixel level requirement and just look at how long it takes for the viewfinder to get still, the damping time is still dependent on the tripod stiffness, torque, and inertia on the load on the tripod.

This formula can be interpreted more intuitively in graphical form:

In red we have the initial displacement caused by whatever torque happens to be acting on the camera. In this case it may be operating the camera’s controls or pressing the shutter button. The green line shows when the camera has settled to a point where any motion is too small to cause blurring in the image. The difference between these two lines is what is given inside the natural log term in the damping time formula above, . The stiffer the tripod used, the initial displacement (red line) will be smaller. With a sufficiently stiff tripod, the red line can drop below the green line and therefore zero damping will actually be required. This is also the scenario we require when we want the camera to remain still, for example, during a long exposure with light winds.

The rate at which the vibration is damping out is shown by the black arrow and given by the ratio of the moment of inertia to the damping constant . The higher the damping constant, the fast the damping occurs, and the more inertial mass placed on the tripod, the slower it occurs. This has an interesting consequence. Minimizing inertial mass is just as important as increasing the damping of the tripod when it comes to settling time. Hold that thought for a future post.

We have slowly been assembling all of the data to solve this damping time equation. Pixel pitch and focal length are just pulled from the camera’s specifications. Tripod stiffness is well understood and measured at this point, and we have some data regarding how much torque is placed on the system. What we don’t have is any measure of the moment of inertia of the cameras and lenses we are asking the tripod to hold. This will be our final necessary piece of information. So, when you see a post in the near future measuring the inertial mass of various equipment, don’t think that this blog has gone completely off the rails into the most boring possible subject. There is a good reason we need to know.

**Takeaways:**

- The amount of time it takes for a tripod to damp out vibrations from an impulse can be calculated in a straightforward manner.
- This time depends on a combination of many factors such as inertial load, tripod stiffness, and focal length in addition to the amount of damping in the tripod.
- Therefore, damping time is a poor way to measure tripod performance in a general way.

**Questions:**

- What are the moments of inertia for typical camera and lens combinations?
- Can we minimize the inertial mass of our existing equipment for better performance?
- What are the calculated damping times for typical shooting scenarios?
- Do these calculated times match up with reality?
- How much tripod damping is necessary?

This rubber cap fits somewhat loosely. So to make sure that I was getting the best possible performance out of the tripod, I removed the rubber caps and tested the tripod with the spikes on the concrete floor. The results was a shockingly low amount of damping in the tripod.

Here is the damping and oscillation curve for the tripod with the rubber caps:

The damping here is poor, but not outside what I have seen in other tripods. Now with the spiked feet:

Note the difference in time scale. The damping is about half, all the way down at 0.082 Js/rad. This is beyond the point of being bad and actually becomes really cool. Nearly half a minute later after the initial excitation, it is still vibrating. Often, designers want as little energy loss in an oscillating system as possible, such as when building a clock. You want the clock to run for a long time without needing to be wound up again. For tripods though, we want the opposite, for those vibrations to damp as quickly as possible.

By removing the rubber feet, we have removed a major source of energy loss from the system. The small amount of rubbing between the spike and the rubber, and the rubber and the floor, was accounting for about half of the damping in the system. The sharp point of the spike directly on the hard concrete floor of my garage removed all of this friction and thus damping. Note thought that the stiffness of the tripod increased slightly from 1308 to 1353 Nm/rad. This increase in stiffness is likely not worth the massive loss in damping observed though.

I don’t mean to imply here that spikes are worse than rubber. The story would probably be very different on a softer surface where they spike could actually embed itself somewhat into the ground. Testing this further is on my to do list, but currently my test setup is confined to the lab, so it won’t happen soon. We can probably safely conclude that spikes are not the best solution for use on firm surfaces.

This also tells us that the feet of a tripod can make a big difference in the damping performance. I didn’t expect this as I have typically seen only marginal differences in the performance of the system based on foot type. Here though, the Jobu Killarney has so little inherent damping to begin with that the damping effect of the feet is amplified. This piques my curiosity as to whether a foot with exceptional damping performance is possible.

**Takeaways:**

- Apparently the choice of foot on a tripod can have a big impact on its damping. This could be an interesting avenue for further research.
- Probably don’t use spikes on hard floors.

**Questions:**

- How much does damping even matter anyways? Working on it, stay tuned
- What are the best feet for different types of surfaces? Concrete? Dirt? Grass? Sand?
- Is it possible to make a foot that adds a lot of damping without any loss in stiffness?