Tripod legs and heads are typically sold separately.  Even when they are not, they almost always connect at the tripod apex via a 3/8″ screw thread.  The logic here is sound.  Photographers and videographers use different sets of gear in different scenarios and therefore need to be able to mix and match equipment.

The stiffness measurements for tripods legs and heads taken on this site are thus done separately.  Testing each combination of head and legs would be totally impossible.  In practice however, tripod legs and heads are used in conjunction as a single system.  We thus need a method for calculating the combined stiffness of the system.  Fortunately, summing the stiffness of springs in series is well understood and fairly straightforward.  The stiffness of the total system can be expressed as the reciprocal sum of the head and leg stiffness’, as follows:

where is the stiffness constant.  A trivial bit of algebra then yields:

In theory this is great and all, but how well does this work in practice?  The stiffness of the head is initially measured by applying the above formula with the head on a very stiff tripod, as described in a previous post.  So, any reader is justified in being skeptical that this will work well in the real world where we aren’t working with ideal springs.  There are a variety of confounding factors such as the myriad of different top platforms on tripods that the heads mate to.

To put the above formula to the test, I measured the yaw and pitch stiffness for four tripod legs and four heads, for a total of 16 different combinations.  I endeavored to choose a range of stiffness’ on each such that we would have weak heads on strong legs, strong heads on weak legs, and ones that were appropriately matched.  I calculated what the above formula suggests the tripod + head stiffness should be, as the ‘theoretical’ result, and compared that to the measured combined system stiffness.  The results are in the table below.  As always, the units of stiffness are in Nm/rad.

Not too bad!  The median magnitude of the error in yaw stiffness is 3% and for pitch stiffness, 7%.  This means that the springs in series formula works quite well here, and will be a useful tool going forwards.  There are some noticeable outliers though, in particular with the pitch stiffness on the Manfrotto BeFree Advanced.  I suspect that the center column locking mechanism on this tripod is subpar.  The additional height of the head then causes the torque placed on the center column lock to magnified when placed under load.  The simple approximation of springs in series clearly breaks down in this case.

Most of the error is on the side of the system being slightly weaker than predicted.  This isn’t too surprising.  When the heads are tested they are secured to the solid aluminum top plate of the underlying tripod with a 3/8″ bolt and a hex wrench from the underside.  This is a stiffer connection than the average tripod can provide.  In addition, when testing the legs, the camera bar is torqued down much harder and and has a larger contact patch than most heads.  Since the real life connection between the legs and the head isn’t as ideal as the test scenarios, we might expect that the stiffness suffers a bit.

There is another approximation that is implicit in the testing but that is working against us in a small way here.  When I test tripod legs and heads, I ignore the moment of inertia of the leg or head itself when calculating the stiffness from the resonance frequency.  This is simply because I don’t have a good way of estimating the inertia of the legs or head as I do with the camera bar and weights used for the test.  I validate this approximation by using vastly more inertia in the camera bar and weights than can be in the tripod or head, and so the resulting errors are <1%.  However, the error is additive when testing the tripod and head together, as is the case here.  So I might expect the actual stiffness’ of the system to be slightly higher than reported here, but only perhaps by 1%.

That we are even talking about errors on the order of 1% is a massive success for this model for combining tripod and head stiffness.  Not only do tripods and heads behave very nearly like ideal torsion springs, but their behavior when adding in series is (with the exceptions discussed above) as expected.  Given the myriad of approximations and error sensitive formulas we have gone trough get to this point, I consider these results a rousing success for the tripod torsion spring model.

Using this model, the clear next step will be to calculate the optimal head for a given tripod in terms of maximizing stiffness / weight.  This has been a long time coming, and I am excited to finally be able to provide a data-backed answer to the question of “Which head should I buy?”