Defining Rotational Equilibrium

Members do not see these Ads. Sign Up.

This is probably one for Kevin Kirby but I welcome answers from any one.
During some research into Rotational Equilibrium, I found the following statement form the Physics department of Winnipeg University
' Rotational Equilibrium is based around rotating RIGID bodies' with the defn ' A rigid body is one that does not deform during its motion. The distance between any two ponts in the rigid body remains fixed'.
Can the application of Rotational equilibrium around the subtalar joint therefore be applicable due to the subtalar joint - the focus of rotational equilbrium theory - not being a rigid body.
I think there is an obvious answer to this but I am strugling to find it.

Any help?

Cheers

Phil

 

Hello Phil,

This subject have been discussed on http://www.podiatry-arena.com/podiatry-forum/showthread.php?t=1521&highlight=rigid lever. The definition you found is correct. Here it lays one of the main problems (at least for me) about Rotational Equilibrium Theory, it is necessary to consider the foot as a rigid lever (rigid body) instaed of mobile adaptor (flexible body).

 

The concept of rotational equilbrium is helpful not just for studying the subtalar joint (Kirby KA: Rotational equilibrium across the subtalar joint axis. JAPMA, 79: 1-14, 1989; Kirby KA: Subtalar joint axis location and rotational equilibrium theory of foot function. JAPMA, 91:465-488, 2001). It can be used and applied to the study of any joint within an animal's body that has an axis of motion.

Let's first get your definitions straight. A rigid body is simply an approximation used for studying the motion of an object where the points on the object are given to not deform relative to each other. This is not real, since all objects deform under loading conditions, however, rigid body mechanics is a whole branch of biomechanics and it is routinely used as an approximation to simplify the study of the motion of the segments of the body.

If two rigid bodies move relative to each other so that there is a rotational motion of one body relative to another, then there must be an axis of rotation between the two rigid bodies for that instant in time where the movement was recorded. This is called an instantaneous axis of rotation.

In the discussion of the subtalar joint (STJ) axis, one rigid body is the talus and the other rigid body is the calcaneus. These are valid approximations since the talus and calcaneus do not deform much except under high loading conditions. When the calcaneus undergoes a rotational motion relative to the talus (eg. the dorsiflexion, eversion and abduction of the calcaneus relative to the talus seen in pronation), then that motion can be shown to occur about a single axis of rotation from the starting position of the calcaneus to the ending position of the calcaneus. This instantaneous axis of rotatoin would then be described as the axis of rotation for the STJ for specific range of motion of the STJ whether is one degree, .001 degrees or 10 degrees of STJ rotational motion. Research has shown that the STJ axis does not have a constant spatial location, but rather moves within space relative to the calcaneus and talus for each degree of rotational motion of the STJ (Van Langelaan, E.J.: A Kinematical Analysis of the Tarsal Joints. Acta Orthop. Scand., 54:Suppl. 204, 135-229, 1983). The numerous instantaneous STJ axes that result have been described as being aligned like the arrows in a quiver, each having a slightly different spatial location relative to the talus and calcaneus, but certainly in the same general location relative to these bones. The STJ Locator relies on the fact that the exit points of these axes from the talus and calcaneus can be approximated since the multiple instantaneous axes of the STJ are probably very close to each other on the exterior surfaces of the talus and calcaneus in most feet (Spooner SK, Kirby KA: The subtalar joint axis locator: A preliminary report. JAPMA, In Press, 2006).

Rotational equilibrium is not a podiatry concept, it is a physics concept that can be easily applied to the study of any joint of the body. It means that the rotational forces (i.e. moments) acting in one direction across an axis of rotation are exactly counterbalanced by the moments acting in the opposite direction. In other words, the summation of moments across the axis of rotation are equal to zero (the moment in one direction will have a positive value and the moment in the opposite direction will have a negative value). This can only occur if there is no acceleration of motion occuring across that that axis of rotation (i.e. either constant velocity or no velocity of rotation). Remember, F=ma so that if the acceleration is zero, then the force (rotational force in this case) must also be zero.

So, rotational equilibrium, as it applies to the STJ axis, means that in order for rotational equilibrium to occur, the STJ pronation moments must exactly be equal to the STJ supination moments so that no net moment is occuring at the STJ (i.e. the STJ is either at rest or is rotating at a uniform velocity). By using these basic physics concepts and applying it to the study of the kinetics of the STJ, the clinician that has a basic background in physics should be able to better understand how orthoses work, why sinus tarsi syndrome occurs, why posterior tibial dysfunction occurs and why peroneal tendinitis/tendinosus occurs.

 

Javier and Colleagues:

There is no inconsistency between rotational equilibrium theory as it applies to the subtalar joint (STJ) and to the long-standing clinical observation that the foot can be both relatively rigid at some times and a mobile adaptor at other times. If the subtalar joint is static, undergoing no motion, then it is in rotational equilibrium, by definition! If there is a net pronation moment, the STJ will accelerate in the pronation direction. If there is a net supination moment, the STJ will accelerate in the supination direction. If you understand the concepts of rigid body mechanics, then the concept of rotational equilbrium of the STJ should provide no inconsistencies when considering the ability of the foot to be both a rigid lever and a mobile adaptor. I suggest that anyone that wants to better understand these concepts purchase this excellent book for your further education on biomechanics: Fundamentals of Biomechanics: Equilibrium, Motion, and Deformation by V.H. Frankel (Foreword), R. Skalak (Foreword), Nihat Özkaya, Margareta Nordin, Dawn L. Leger (Editor)http://www.amazon.com/gp/product/03...104-6868649-3269508?s=books&v=glance&n=283155

 

Phil, Javier and Colleagues:

By the way, static equilbrium is a special type of rotational equilibrium where the rotational velocity=0.

Here are some more websites on rotational equilibrium, static equilibrium, rigid bodies, moments, torques, etc. I just wanted you all to see that rotational equilibrium is standardly taught in physics classes around the world....I think it is about time that the concept of rotational equilbrium is also standardly taught in podiatry schools around the world so that this standard physics concept of rotational equilibrium is not such a new thing to podiatry students and podiatrists now 17 years after my first paper on the subject of rotational equilibrium in the subtalar joint in the podiatric literature was published.

What more do I need to do before podiatrists will start realizing the importance of rotational equilibrium for their understanding of the biomechanical function of the foot!?!? Die an early death????!!!!

http://www.ux1.eiu.edu/~cfjz/SP0511510309.pdf#search='rotational equilibrium, definition'

http://www.physics.rutgers.edu/ugrad/205/torques.html

http://www.phy.cmich.edu/people/andy/physics110/book/Chapters/Chapter7.htm

http://www.physics.csbsju.edu/RPEG/no_paper/handouts/Lesson.08.html

http://www.physics.umd.edu/courses/...f#search='rotational equilibrium, definition'

http://dept.physics.upenn.edu/courses/gladney/phys1/lectures/lecture9/phy009_pg2.html

http://physics.bgsu.edu/~stoner/p201/equil2/

http://www.cdli.ca/courses/phys3204/unit01/section04/lesson01/3-lesson-a.htm

http://students.cup.edu/mcc8760/

 

Hi Kevin

When I studied Podiatry (this is only my 4th year since graduating) the curriculum included Physics subjects, and Biomechanics subjects, with the physics mostly including basic principles of forces, torques, fluid dynamics and movement, and the biomechanics being largely Root based teachings.

As I have mentioned previously, the concept of applying basic physics principles of force and moment to the subtalar joint model was quite foreign to me until you posted the Thought Experiments here. I agree that it would be most valuable for Podiatry students to learn the Subtalar Joint Axis Rotational Equilibrium Theory and the Tissue Stress Theory, especially since the mechanical principles ( IMO) are much easier to understand when explained in this way.

Regards

Donna

 

....a Kirby Memorial Seminar!!

I would hate to think how many times in the last few weeks of lecturing students I have said something like "thats why you got taught all that stuff in first year". It been almost 8 years since we have been teaching students sagittal plane, tissue stress, planal dominace etc etc and also make links between all that and what they got taught in first year re forces, moments etc etc.

Don't you find it problematic that in so many podiatry schools that they get all the forces, moments etc in first year, but no attempt is made in what I prefer to call "clinical biomechanics" to integrate and further develop that earlier stuff. Why bother teaching any basic sicence if its not developed further in clinical subjects?

We are due to have another major think through just how and what we teach in this area to develop it even further.

 

Phil

The assumption that a body (the foot for instance) is rigid for the purposes of analysis is quite valid.
Here is why:
1) If you do not assume a rigid body then this becomes a problem of finite element analysis. This means that you must consider the deformation of a material, or several materials in the body of interest, under stress. This becomes virtually impossible without the use of computers and a lot of time and expertise. Finite element analysis must be used when there are more unknowns than there are equations to solve them using rigid body or static/dynamic equilibrium principles.

This FEA technique can be used with confidence in engineering, and is, since values of stiffness (youngs modulus etc) of construction materials are well documented and programed into software analysis products.
A major problem with using this technique in analysing human kinetics and kinematics is that the strain/stiffness values of each tissue are poorly defined and research produces widely varying values. There are obvious difficulties in making in vivo studies and in vitro studies are almost invalidated by that fact.
The amount that a certain tissue strain value varies inter subject or even intra subject is also almost unresearched as far as I know.

There are software products such as BRG LifeMod and AnyBody technology, which use finite element analysis to produce virtual models of an activity.
How accurate these are I don't know but they are very interesting.

2) This is what makes this rigid body technique valid.
To analyse the kinetics dynamically first one must have some known values. These values can be force applied to the foot. If these forces are measured the sample taken is at a finite or instantaneous point in time. At that point in time the sum of all the variables of strain in each tissue are equal to one rigid body that has the equivalent value of stiffness.
If one has a complete data set of such known values, eg force applied to the foot over a certain time, then it is possible to make conclusions, using a graph for instance, about the value of the unknowns of interest eg the moments about the STJ. Interpolation of these results would be accurate if the sampling is at a high enough frequency IE sample per second.
Static analysis has the same principle.

The major point to remember here is Newtons third law, 'every force has an equal and opposite force', In any situation static or dynamic there is always equilibrium of forces.

Example. A see saw (The good old see saw analogy) has a plank and a fulcrum. The fulcrum is rigid (for our purposes) the plank is not. This does not mean it is not possible to find the moments about the fulcrum.
The see saw is in motion. The plank can flex under load. However at the point in time when the force applied to the see saw, at distance L, is measured it can be considered to be a rigid body and the the moments calculated as Force x Lever arm. In this case we would need to measure the lever arm length which would change as the plank flexed.
In the human body the deformations at the foot often can be considered to be negligible but if one required more accuracy then it would be neccesary to record the changes here also. (to look at the STJ in relation to calcaneal frontal plane displacement for instance).

Is this clear to you Phil?

Cheers Dave

 

Here's some lecture slides that an orthosis lab has used from my papers and lectures for a presentation they have developed on subtalar joint axis location and rotational equilibrium: http://www.parisorthotics.com/hcp_lectures/STJA Loc and Rot Equil Theory_PAC2001.pdf

However, our 1988 study (Kirby KA, Loendorf AJ, Gregorio R: Anterior axial projection of the foot. JAPMA, 78: 159-170, 1988) did not show that one of the primary weightbearing structures was the 5th metatarsal head, as they stated in one of their slides. We did show that the primary weightbearing structure of the plantar heel was the medial calcaneal tubercle and that the lateral calcaneal tubercle probably never became weightbearing even in extreme inversion in most feet. This is quite different than what is illustrated all through Root et al's textbooks and Sgarlato's compendium which, I believe, gives the reader a wrong impression as to how subtalar joint stability is maintained during standing and weightbearing activities.

 

No discussion here. Rotational equilibrium theory fits with STJ (considering its anatomical and mechanical characteristics). When I refered to dicotomy rigid body/mobile adaptor, I was thinking about whole foot (not only STJ). As It was discussed in another thread, rotational equilibrium can not used (at least for now) to midtarsal joint.

It should be taught along with biophysics (theorical and practical). It would help to leave many theorical nonsense.

I am afraid that it is only useful for rock stars!

 

Hi Craig,

I agree with what you are saying, the basic science that we are taught in 1st and 2nd year needs to be integrated so that a deeper understanding can be developed.

The Thought Experiments are a good example of this... I was fairly rusty on my physics when I attempted these problems, and I would imagine that a 2nd year student (who had more recently completed the physics unit) would have them completed much more quickly. You mentioned that you actually present some of these types of problems to your class, and I can see how the students would find this most valuable as it reinforces the basics by applying concepts to an easy to visualise model.

I'm glad that Podiatry Arena is available, I am constantly learning from the Senior members, as they put such a lot of time and effort into posting on this forum. It's like having a classroom inside the computer!

Regards

Donna

 

Every biomechanics lecture I have given recently to the students lately has involved refering them to at least a couple of thread here related to the lecture - its helps them gain a better understanding of the topic and often presents the subject matter in a different way to I did. They especially like the threads where people disagree with me.
The thread on Bojsen-Mollor was particularly useful to them (and...several disagreed with me )

 

I just wish I'd known about Podiatry Arena earlier! But I am making up for lost time!

Regards

Donna

 

Happy to be able to contribute to the contentment of your students, Craig.

 

Javier:

I have recently been writing Precision Intricast Newsletters on how the foot can be rigid at times while flexible at other times. I do not include a discussion of the subtalar joint within these Newsletters in order to explain this seemingly dichotomous functional characteristic of the foot during weightbearing activities.

In addition, I have written and lectured numerous times on the concept of midtarsal joint rotational equilibrium to discuss the following:

1. How foot orthoses work
2. How medial longitudinal arch flattening causes functional hallux limitus
3. How the longitudinal arch of the foot maintains its shape during weightbearing activities
4. Why posterior tibial dysfunction causes abduction of the forefoot on the rearfoot.

While I have yet to write a definitive paper on this subject, midtarsal joint mechanics is one of my favorite current subjects. It will also be the subject of my lecture at the PFOLA meeting in Chicago in early December 2006 (Chris Nester will also be presenting with me in the same session on his work on the midtarsal-midfoot joints).

Therefore, midtarsal joint rotational equilbrium is as important a concept and deserves just as much attention as subtalar joint rotational equilibrium.

 

Thanks for everyones input on this topic.
Just to be clear, I believe that rotational Equilibrium is, along with H Dannaberg Sagitall plane mechanics and E Fullers Tissue Stress approach, the closest thing to real biomechanics that we currently have in podiatry/musculoskeletal medicine. Like most approaches it is very simple, easy to understand and clinically relevant when used in conjunction with other approaches and not taken in isolation. (But again thanks for the simplified and clear explanations, as it ensures that those people who do not understand this fundamental approach should do by now)

The reason for the question was due to the medico-legal aspects of orthoses provision and education( Something that I do in my job every day). I was taught by Tony Redmond at an undergraduate level (Hudedrsfield Univesity) to question EVERYTHING and not assume anything. It was with this in mind that I am currently de-constructing everything that I practice to ensure that the advice I give in my job is as robust and evidence based as possible.

You have all managed to put my mind at rest that the risk of litigation when teaching RT is virtually non-existent.
Now on to the next attempt at de-construction.

Cheers

Phil

 

I agree with you. The theories you have mentioned are paradigms http://en.wikipedia.org/wiki/Paradigm . Sometimes, paradigms are considered as something unchangeable and infallible, not such a therorical model for explaining reality.

As it was discussed on the thread http://www.podiatry-arena.com/podiatry-forum/showthread.php?t=1521; Midtarsal Joint equilibrium theory through a single axis is a nice and clean paradigm; but not enough to produce a "paradigm shift", because medial longitudinal arch behaviour to weight-bearing can be explained by other ways.

 

Kevin
Thanks for the great info and web links. I have another question if you don't mind.
You speak about the STJ axes projecting out along the foot. Are there any circumstances where these projections can be effected enough to be invalid. I am thinking clinically that an excessivley laterally deviated forefoot or rigid plantarflxed 1st ray may create a retrogarde force on the stj that may only happen at a certain phase in gait. Also mechanically speaking, can a foot segment that is excessively flexible or less stiff allow too much translation to now let the rigid model be applied.
If this is sound reasoning, do you have any clinical examples of where we need to take this into consideration?

Thanks again

Phil

 

Phil:

These are excellent and insightful questions that you have. First of all, the rigid body model that I used in my papers on subtalar (STJ) axis location (Kirby KA: Methods for determination of positional variations in the subtalar joint axis. JAPMA, 77: 228-234, 1987; Kirby KA: Rotational equilibrium across the subtalar joint axis. JAPMA, 79: 1-14, 1989; Kirby KA: Subtalar joint axis location and rotational equilibrium theory of foot function. JAPMA, 91:465-488, 2001) is a standard approximation used in research on foot and lower extremity biomechanics. Of course, the foot moves and deforms under weightbearing loads, however, this does not mean that the rigid body model can't be used to approximate the moments acting across the STJ. As long as the foot can be viewed as being relatively immobile over an instant of time, then the rigid body approximation will be very close for weightbearing moments acting across the STJ axis.

In your example of a foot with a plantarflexed first ray that may show a significant change in position during gait, the rigid body approximation can still be used to approximate the STJ moments at any instant of gait. In order to find these STJ moments, you must be able to determine three parameters:

1. Determine the plantar location of the CoP at any time during gait (e.g. by force plate or pressure mat or pressure insole)

2. Determine the magnitude and three dimensional direction of the ground reaction force vector (e.g. by force plate)

3. Determine the spatial location of the STJ axis during gait, which can only currently be approximated by use of the STJ axis locator (Spooner SK, Kirby KA: The subtalar joint axis locator: A preliminary report. JAPMA, In Press, 2006) but will hopefully soon be able to be tracked in real time using a 3D motion analysis system (Lewis GS, Kirby KA, Piazza SJ: Determination of subtalar joint axis location by restriction of talocrural joint motion. Gait and Posture. In press. 2006).

This type of "quasi-static" method of kinetic analysis, in other words, breaking down the stance phase into incrementally small periods of time to determine the STJ moments at any instant in gait can be used as long as the accelerations of the foot are small [foot accelerations are almost always very small during the stance phase of gait]. This quasi-static determination of joint moments can therefore be used as a valid approximation of STJ moments during the stance phase gait in nearly all cases.

Here is a good discussion on Biomech-L list on the subject of inverse dynamics and quasi-static modelling http://isb.ri.ccf.org/biomch-l/archives/biomch-l-2002-01/00021.html

Also, page 62 of this chapter discusses quasi-static determination of joint moments http://www.vard.org/mono/gait/soutas-little.pdf

No matter how flexible the foot, the CoP, GRF vector and STJ axis spatial location can be used during stance phase to detemine STJ moments. The problem is that we don't really know where the STJ axis is within space at all times during gait and we can't find the GRF vector without a force plate. So, until that time, we will need to use an educated guess at what the STJ moments are in each foot we examine and basically just try "to push the foot away" from the position that causes injury with our foot orthoses until we possess the technology that allows us to have more concrete information on the direction and magnitudes of STJ moments during gait in our patients.

 

Phil

If you consider a compliant r/foot 1st ray V's a stiff l/foot 1st Ray then you could attribute stiffness values of force/deflection this could be 50N/mm compliant r/f and 100N/mm stiff l/f. As they contact the ground the and the full force of body weight is applied (for arguments sake just on the 1st rays) then if the b/w force is 400N on each ray then the stiff ray will flex by 4mm and the and the compliant one by 8mm.
So if at the point in time that each one is deflected by half its max range for this b/w what will be the force under the met head? 200N each.
By knowing the force applied to the met head and its location relative to the axis of interest you can work out the Moments as a rigid body static analysis.
Inertial effects are negligible in this case as the ray has a small mass and does not accelerate very quickly. If you were to do the same for the moments at the hip then you would need to look at inertial forces. This is done by finding the body parameters for the leg of interest, dimensions, CoM position and magnitude, radius of gyration, acceleration, moment of inertia, these values plus the moment due to grf will give the external moments about the hip. Because the individual joints of the limb can be considered to have a certain degree of stiffness at a certain point in time (just like the 1st ray example) all the segments of the limb are considered rigid at the point of analysis.
The mass of the foot or its segments can be considered negligible and so finding these moments is simplified. Both example are analysed by the static rigid body model.

Dave

 

Dave
I agree with you are saying but as Kevin mentioned in his last posting re. the ability to define the STj at any instance is dependant on GRF vectors, CoP etc. If the 1st ray is fixed and therefore dosn't yield to GRF, then the medial column has the potential to act as a lever, with its fulcrum being at the STj. (I think the term to best describe the combined action of the midtarsal joints would be a series of coupled levers/moments- not totally sure on this one though) This potential rotational force i.e. an inversion moment, could result in the alignment of the STj being effected. If in the clinical situation the forefoot was also plantarflexed in a fixed and rigid position, the GRF would be different around the STJ.
The question would then be ' do changes in maginitude of the GRF e.g 'shared' by the rearfoot and the forefoot at initial contact, change the compression of the joint and consequently its axis?

Kevin, thanks again for the links and further descriptions

Phil

 

That is not exactly what I said. The determination of STJ axis spatial location is independent of determining the CoP or the direction and magnitude of the GRF vector. However, the determination STJ moments is dependent on determination of STJ axis spatial location, CoP location and direction and magnitude of the GRF vector.

The first ray is never "fixed", it always moves in response to GRF. Please don't use the term "fixed" since it implies the inability to be moved. It is better to say that the first ray has increased dorsiflexion stiffness, so that when it is loaded by GRF, it dorsiflexes relatively little (Kirby KA, Roukis TS: Precise naming aids dorsiflexion stiffness diagnosis. Biomechanics, 12 (7): 55-62, 2005). In addition, whether the first ray is stiff or, instead, is more compliant to dorsiflexion loading forces, the first ray will always be able to transmit GRF acting on its plantar aspect to cause rotational forces (i.e. moments) across the STJ.

However, with increased first ray dorsiflexion stiffness, the first metatarsal is likely to have increased magnitudes of GRF so that the CoP will likely also be shifted more medially in the forefoot. This medial shift in forefoot CoP would either increase the magnitude of STJ supination moment or decrease the magnitude of STJ pronation moment which may, in turn, cause STJ supination motion which, in turn, would cause lateral translation and abduction of the STJ axis relative to the plantar forefoot. This shfting to the STJ axis in space with STJ supination and pronation motions is a very important concept when considering the STJ axis spatial location at any time during gait (Kirby KA: Subtalar joint axis location and rotational equilibrium theory of foot function. JAPMA, 91:465-488, 2001). Unfortunately, this fact is only appreciated by a small fraction of podiatrists.

The GRF acting on the plantar first metatarsal head is transmitted instantaneously through the more proximal bones and soft tissues of the foot to produce STJ moments. This is because the foot behaves as a rigid body if the midfoot and midtarsal joints are undergoing low accelerations (i.e. are moving little) in response to the GRF plantar to the first metatarsal head.

The sharing of load by the rearfoot and forefoot are taken into account by the center of pressure (CoP) parameter. CoP can be measured by pressure-sensing insoles, pressure-mats and force plates. If less GRF is present on the forefoot and more GRF is on the rearfoot as in early stance, then the CoP will be relatively posteriorly located. If less GRF is present on the rearfoot and more GRF is on the forefoot as in late stance, then the CoP will be relatively anteriorly located.

The compression of joint surfaces is more a function of external forces, such as GRF, being converted into internal forces, such as ligamentous tensile forces and joint compression forces. The magnitude of joint compression forces is dependent on many other factors, beyond the scope of this discussion.

 

Rotational equilibrium useful approach

Here it is a case where Rotational equilibrium is useful because the foot acts as a rigid body. When a transmetatarsal amputation is performed, the stump usually works as a rigid body and lateral or medial displacement from the CoP can bring to an ulceration on the distal part from the stump. Thus, forefoot plays some role on the foot function, does not it?

I attach some photos below (I hope it works).

 

photos from the previous e.mail

I attach the photos from the previous e.mail

 

Thanks a lot to the above threaders, i suggest when we make these debates, it is vital that we hold certain facts constant so that we don't have too many variables, that will end up distorting the argumentative facts. Why do i have a feeling with the foot and ankle in particular we are trying so hard to stick facts to it to justify our complicated debatable understating (yet we have so many variables that are constantly challenging our so called paradigms?) of its function??. boy oh boy we try so hard and i am grateful for it, because what we are doing is what we ought to have done i guess. i have spoken to a few physicists and the moment you introduce the feet, some just get overwhelmed and their principles seem to fail to narrow done that amount of variables for a possible fact to hold esteem why??? i can see kevin pulling his hair lol?. So stj neutral position does exists???? yes or no!