CGA FAQ: Centrifugal force & inertial frames

"Dr. Chris Kirtley" wrote:

 > Incidentally, I've always wondered why there are no centrifugal forces
 > included David Winter's 2D inverse dynamics analysis (in Biomechanics
 > and motor control of human movement and elsewhere). Did David leave
 > these out because they are negligible in gait, or for soem other reason?

 I think the centrifugal force is not included in these equations because
 the equations of motion were written for motion measured in an inertial
 reference frame.  Centrifugal force only appears in equations of motion
 written for movement in a rotating coordinate system.

 Knowing that Stalin did not allow non-inertial reference frames (thanks
 to Arnold Mitnitski for this interesting piece of information), I can't
 resist offering a few examples where using non-inertial frames seems to be
 a good way to do the calculations.

 Example 1: Weather forecasting is done by solving large finite element
 models on a mesh that is attached to the earth.  And since the earth is
 not an inertial reference frame, Coriolis forces and centrifugal forces
 (the latter are probably insignificant) must be added to the equations.
 This does not make the calculations more difficult; these "pseudo-forces"
 are very well known.  One advantage of this is that it makes things
 easier to understand, for instance why the Coriolis force makes hurricanes
 spin counterclockwise in the northern hemisphere.  But the main reason is
 convenience in the computational work.  Even though it is true that the
 solution would be the same when the equations are solved in an inertial
 frame, one can imagine the difficulties when weather forecasting would be
 done on a mesh attached to the sun, or the galaxy, or the center of mass
 of the universe...

 Example 2: Some years ago I was involved in a study on inverse dynamic
 analysis of downhill skiing.  Because of the large volume needed for
 movement analysis, we considered using a system to measure only the motion
 of the body segments relative to the boot, definitely a non-inertial frame.
 When transforming the equations of motion to this reference frame, "pseudo-force"
 terms appear that include the state of acceleration (linear acceleration,
 angular acceleration, and angular velocity) and orientation of the reference
 frame relative to the earth.  It also appeared that these terms could be
 determined from a number of accelerometers rigidly attached to the
 non-inertial frame.  So, inverse dynamics can theoretically be done in a
 non-inertial frame with a completely body-mounted instrumentation system.
 In this case, transforming all motion data to an inertial reference frame
 is not even possible, because we don't know the motion of the non-inertial
 frame.  We only know its state of acceleration.  We did the project somewhat
 differently in the end, but at least I know that it is theoretically possible
 and that it requires equations of motion to be written for the non-inertial
 frame. And those equations include pseudo-force terms.  I don't think this
 type of analysis can be done in an inertial reference frame.

 In both cases, I guess the reason for using a non-inertial frame is the
 difficulty of collecting movement data relative to an inertial frame.  It's
 fine to write the equations in an inertial frame, but what if you don't have
 the data that is needed to do something with the equations?

 Finally, I fully agree with Chris Kirtley mentioning Einstein's principle of
 general relativity.  According to that principle, there is no way of knowing
 whether a force that we measure (e.g. gravity) is "real", or "just a pseudo-force"
 which is a consequence of doing measurements in a non-inertial reference frame.
 General relativity treats gravity as a pseudo-force just like the centrifugal
 force.  Even Stalin would agree that gravity belongs in a free body diagram,
 but in fact gravity is no more "real" than a centrifugal force.

 Ton van den Bogert

 P.S. For an explanation of the effect of Coriolis forces on the weather,
 and some critical comments on draining sinks, see

 For an introduction to general relativity, see


 A.J. (Ton) van den Bogert, PhD
 Department of Biomedical Engineering
 Cleveland Clinic Foundation
 9500 Euclid Avenue (ND-20)
 Cleveland, OH 44195, USA
 Phone/Fax: (216) 444-5566/9198

Dear all and everybody,

 Probably because of my laziness I did not go back through the BIOMECH
 archives to a discussion in March, 1999 on this subject before writing
 this letter, though I think that Prof. Gottlieb gave a good advice.  If
 I (and not only me, as I can see) did not find time to look at this
 materials readily available with a click of the mouse, why to be
 surprised that somebody did not have time (or energy, not necessarily
 kinetic one, though it is not excluded) to read some good books in
 mechanics, specifically the chapters devoted to two basic concepts:
 "inertial frame of reference" and  "relative motion".  Theoretical
 mechanics (sometimes called "classical")  had been developed a while ago
 and good OLD books have all the information necessary to understand the
 concepts as well as to apply them. I've got an impression that
 contemporary manuals (I did see some of them) are very much biased
 towards the applications. I am not complaining about that, at all, but
 it seems that "public opinion" is to leave the Theory and basic concepts
 for Physics.  May be it is possible to design bridges, mechanisms and
 even robots without such concepts (I tempted to say, without
 understanding what the laws of Newton are all about, but I don't say
 that!) especially if any software is available. I am not so sure that
 for satellite dynamics it is enough, however. But Space problems are not
 the most important yet, at least for biomechanics.  Trying to finish
 this message with a more optimistic tone, I am thinking what book may be
 recommended to get insight into the concepts of Inertial Systems and
 Relative Movement.  I know some excellent books in Russian (may the best
 is classical "Physical Foundations of Mechanics", by Khaikin published
 sometimes around 50s) bit I don't know if English translation is
 available.  You will probably find it curious that during Stalin, it was
 a noisy discussion in Russia about inertial forces, they were condemned
 to be the wrong concepts and the scientists who were not very careful in
 introducing them to the students could be considered as the ideological
 enemies and could be persecuted (some of them were, and D'Alambert would
 be the first if he was alive). What I like about the present discussion
 is that nobody will be persecuted (either feel offended, I hope) and may
 voice any opinion openly and freely.  Is it not a triumph of our
 democracy?  Happy Chanukah and Merry Christmas!

  Arnold B. Mitnitski

  Ecole Polytechnique, Applied Mechanics Dept. P.O. Box 6079,
  Station "centre- ville",  Montreal, PQ, H3C 3A7, Canada
  Tel.:(514) 340-4711 X-4861;  Fax: (514) 340-4176

Dear Biomch-l discussiants,
 Funny that this topic immedeately gives such a strong response!
 Although all sensible points have been raised already, I cannot but
 add my own viewpoint.
 In fact both those pro- and contra virtual forces have their point:
 1) These virtual forces are not needed when you stick to an inertial
 reference frame. That is why they are not in the book af Winter. His
 analysis is based op optokinetic recordings. It is then natural to refer to an inertial
 laboratory frame.
 2) They can come in very handy in some calculations.

 My point is: how to teach this to students? My experience is that
 they can quite easily learn to do the calculations, but  some
 simple insights are very difficult. A main point is Newton's third
 law, and the concept of a free body diagram. On what body is the
 force working? Most students are very inclined to draw both the force
 on the body and the force from the free body on the outside world (to
 happily conclude that both are equal and opposite and thus cancel).
 This is why I very carefully avoid to introduce 'virtual forces',
 because I expect that would increase confusion to an all-times high.
 The best is thus to stick to a laboratory frame of reference, and
 give some arguments why it is to be preferred.
 All the same, d'Alembert's principle can be very handy. It gives you
 the immedeate solution of the moment equation for a set of coupled
 rigid bodies. I found this out some years ago, J. Biomechanics 25:
 1209-1211, 1992. (Interestingly,  neither I nor my reviewers saw at
 the time  that it was in fact a formulation of 'Alembert's principle
 of 1740.) In short:
 For a static system we have the equilibrium equations:
 sum(F) = 0
 sum(M) = 0, around an arbitrary point
 According to d'Alembert for the dynamic case this becomes only
 slightly more complicated:
 sum(F) = sum(ma)
 sum(M) = sum(r x ma) + sum( I*alpha)
         moments again around an ARBITRARY point.
 I try to teach this to the students. It is not easy, but at least
 somewhat more conveniently arranged than the Newton-Euler approach,
 going from segment to segment, and with moments always around the
 centres of mass.
 The entries at the left hand side of these equations are 'real'
 forces and moments. Those at the right hand side I just call 'terms
 ma , r x ma, and I*alpha', never suggesting that they, or their
 opposites, are real forces or moments.
 I wonder whether this approach will allways work, even in the case of

 Ton's meteorologic problems.
 But centrifugal forces... no way.
 Best wishes,
 At Hof
 Department of Medical Physiology &
 Laboratory of Human Movement Analysis AZG
 University of Groningen
 A. Deusinglaan 1,  room 769

 PO Box 196
 Tel:   (31) 50 3632645
 Fax:   (31) 50 3632751

"Dr. Chris Kirtley" wrote:  > As far as I know, we have no sensors for segment acceleration - only  > (conceivably) joint angular acceleration, via spindles, joint afferents  > and skin receptors. Would this variable be sufficient, I wonder, for the  > CNS to compute the inverse dynamics?  In principle, yes, I think.  With eyes closed, we have no information  about our motion relative to an inertial reference frame.  But we  have a set of "accelerometers" in one of our rigid body segments,  the vestibular system in the head.  Then we have sensory information about  relative motion of all our other body segments relative to the head.  If the CNS wanted to compute inverse dynamics in a reference frame  attached to the head, it could, theoretically.  It is another question  whether this is possible in practice, considering the errors in the  sensory signals and errors introduced by the computation in neural  circuits.  Ton van den Bogert  --  A.J. (Ton) van den Bogert, PhD  Department of Biomedical Engineering  Cleveland Clinic Foundation  9500 Euclid Avenue (ND-20)  Cleveland, OH 44195, USA  Phone/Fax: (216) 444-5566/9198
Dear subscribers,  I thank Paolo de Leva for his challenges.  Out of such discussions always  comes more understanding.  First of all, I want to emphasize that there are  practical issues, not just the philosophical ones.  Equations can be used for  two purposes: educational and technical.  For education, it is important that  equations help us understand.  In technical problems, equations provide  relationships between things we can measure and the unknowns that we really  are interested in but can't measure.  From that point of view, if nothing  in an equation can be measured, it is not a useful equation.  This is what  I see as the greatest problem in insisting on using only inertial reference  frames.  Feldman's comments are also important.  The human sensory system mostly does  not have an inertial reference (vision can be inertial referenced) and yet  we can use it control complex movements.  Obviously the required sensory  information is available.  Paolo de Leva wrote:  >     So, unfortunately, you can't just fust forget the inertial (Newtonian)  > frame when you use D'Alembert's principle. It might seem that Ton van den  > Bogert, using "accelerometers rigidly attached to the non-inertial frame" as  > described in his latest message, could deny my previous statement:  >  > > When transforming the equations of motion to this reference frame,  > > "pseudo-force" terms appear that include the state of acceleration...  > >[omissis]... and orientation of the reference frame relative to the earth.  > > It also appeared that these terms could be  > > determined from a number of accelerometers rigidly attached to the  > > non-inertial frame.  >  >    Notice that Ton clearly wrote that he needed and obtained the orientation  > of an inertial frame (the earth is quasi-inertial, but we can neglect in  > this case the effects of its relatively slow rotation).  I apologize for not being more clear, but my point was that you do *not* need  that orientation, and also that you really *can* forget about the inertial  frame, as long as you know those extra terms in the equations of motion.  One of those terms in the equations of motion written for a non-inertial  reference frame is a term due to gravity and acceleration of the origin of the  reference frame.  But the beauty of this is that these two effects always  are combined into one term that can be measured with accelerometers.  Intuitively, this should make sense: body segments "feel" the same forces  that the mass inside an accelerometer feels.  And it does not matter  if that "feeling" comes from gravity or from an accelerating reference  frame.  And Einstein says you can't distinguish between those anyway.  To show this mathematically, let's start with the familiar equation of motion  for a particle in an inertial reference frame:  (1)   F + m*g = m*A  where F = (Fx,Fy,Fz)' represents the sum of all forces except gravity, g =  (0, 0, -9.81) m/s2 and A = (Ax,Ay,Az)'.  The symbol ' indicates transpose, so  these are column vectors.  Now let Fm and Am be the same variables but  measured in a moving reference frame.  For simplicity we assume that  angular velocity and acceleration can be neglected (for the full equations  see my article in J Biomech 29:949-954, 1996).  Let R be a rotation matrix  describing the orientation of the moving reference frame relative to the  inertial reference frame, and Ao be the acceleration of the origin of the  moving reference frame measured in the inertial reference frame.  The relationship between F and Fm is simply a rotation of the reference  frame: F = R*Fm.  The relationship between A and Am can be derived by  twice differentiating the rigid body transformation for coordinates of  a point P: P = T + R*Pm, where T is the translation and R is the rotation  of the reference frame.  In the absence of angular velocity and angular  acceleration, the result is: A = Ao + Ro*Am.  Substituting these into  equation (1) gives:       R*Fm + m*g = m*(Ao + R.Am)  Pre-multiplying by the inverse of R gives:  (2)   Fm + inv(R)*m*(g - Ao) = m*Am  Now this looks exactly like an equation of motion for a particle  again, but there are two differences.  First, gravity has been removed.  Second, there is an extra "force" on the left-hand side which  depends on orientation R and acceleration Ao of the moving reference  frame.  But, and this is important, we never need to know R and Ao!  We only need to know the combination inv(R)*m*(g-Ao).  And this  quantity can be measured with an accelerometer.  So for example, if  your reference frame were accelerating towards the center of the earth  with an acceleration of 2g, you would "feel" exactly the same as if the  reference frame had simply been turned upside down.  In both cases you  would be pulled towards the ceiling by a 1g force field.  And you never  need to know what is really happening, as long as you measure the force  field.  Until you start considering that in one of the two scenarios  you will hit the ground sooner or later :-)  >     Notice, also, that the accelerations measured by the accelerometers are  > observed from an inertial frame. This might not seem obvious,  > but it is absolutely true, in my opinion. The inertial frame used by the  > accelerometers is, of course, tilted relative to the usual horizontal and  > vertical axes of the frame attached to the earth, and its  > orientation changes with time. However, since the accelerometer senses true  > accelerations and not imaginary ones, in a particular instant when you  It is important to realize that accelerometers are sensitive to acceleration  and to gravity.  It is simply a mass attached to a little force transducer.  And  the XYZ components of the signal are measured along axes fixed in the moving  reference frame.  Yes, it is the acceleration relative to the inertial frame,  but it is always combined with gravity and rotated to the moving frame.  An  accelerometer measures inv(R)*(g-Ao) and this is all you need to know.  Again  I am not considering terms related to angular velocity and angular acceleration,  which you can find elsewhere.  The principle stays the same.  >     Here's how Ton concludes:  >  > > So, inverse dynamics can theoretically be done in a  > > non-inertial frame  >  >     Here Ton seemed to say he didn't need the inertial "Newtonian" frame  > (the earth) at all,  > although he just stated above he did. (What did you mean, Ton?). This  > conclusion might be misleading for those who will read it too quickly. And I  > think it is crucial, in this particular  > discussion, not to be mislead in that direction.  What I meant is that you do not need the Earth or any other inertial  frame, as long as you use accelerometers to determine the force field  due to the reference frame being non-inertial.  This requires four triaxial  accelerometers in the general case.  >     Of course, I am not saying that non-inertial frames are useless. I am  > just saying inertial frames are necessary.  And here we disagree.  In fact, I would say non-inertial frames are needed  because usually we can only collect data in a non-inertial frame, and we can't  transform the measured variables to a noninertial frame.  The Earth is a good  example.  It is a non-inertial reference frame and yet we measure forces and  accelerations with our video cameras and force plates in a coordinate system  attached to the Earth.  It just happens that the pseudo-force terms are too small  to have an influence on human movement, so we can get away with ignoring them  in our equations.  But it is important to know that you don't *have* to ignore  them!  That immediately opens up some interesting possibilities.  For instance,  you can do inverse dynamic analysis on a ship, no matter how wild its motion is,  provided that you attach four accelerometers to the ship and include that  information in the analysis.  People who measure and predict weather patterns  have no choice.  They always collect movement data relative to the Earth, but  they *need* to add the pseudo-forces to the equations because they can be as  large as the other forces.  Yes, they could write the equations for some  inertial frame, and these equations would be very simple, but they could never  collect the data to actually use those equations.  Ton van den Bogert  --  A.J. (Ton) van den Bogert, PhD  Department of Biomedical Engineering  Cleveland Clinic Foundation  9500 Euclid Avenue (ND-20)  Cleveland, OH 44195, USA  Phone/Fax: (216) 444-5566/9198
Paolo de Leva wrote:  >     My point was clear and simple (in my honest opinion, of course). Briefly: accelerometers do  > use inertial frames.  We still disagree on this.  In my opinion, accelerometers do not measure acceleration,  they measure force.  And they do not show how much of that force was produced by  acceleration and how much by gravity.  And my point is that we do not need to know,  for the purpose of doing dynamics calculations in a non-inertial reference frame.  > Isn't it obvious that an accelerometer has zero acceleration in its own local frame?"...  The force vector that is measured by an accelerometer indeed can be written as a  function of its acceleration in the global inertial reference frame, and gravity:     F = m.a - m.g  F, a, and g are all 3x1 column matrices expressed in the global reference frame.  However, an accelerometer rotates with the frame it is attached on, so the XYZ  components of that force vector are determined along the XYZ axis of the local frame.  So the three signals we get are Fm = (Fmx, Fmy, Fmz), which can be written as:  (1)   Fm = inv(R)*F = inv(R)*m*(a-g)  This is proportional to the pseudo-force that we needed to add to the equation of motion  for a particle in the moving reference frame.  So this shows that the accelerometer  gives sufficient information to complete the equation of motion.  This does not include  the effects of a rotating frame, since R is assumed to be constant, but terms  related to angular velocity and angular acceleration can be added too.  Now read the following carefully:  What may be confusing here is that we have a 3-D acceleration relative to the  global frame, but quantify that acceleration vector by its XYZ components in the  local frame.  This is a technical necessity: the sensors are attached to the  moving frame, even though the force vector may be the result of an acceleration  relative to the global frame.  To prevent further confusion, let me make it clear that the symbol Am that I used in  my previous posting refers to the acceleration of a particle relative to the local frame,  expressed as XYZ components relative to that local frame.  For an accelerometer, Am is  zero, because it does not move in the local frame.  Equation (1) above can be derived  as a special case of that general equation of motion by setting Am = 0.  So this should resolve that paradox: an accelerometer has zero acceleration in its  local frame, but its signal is not zero.  Its signal is a combination of *global*  acceleration and gravity, expressed as XYZ components along the *local* axes.  This is only a rotation of the reference frame, the magnitude of the vector  is not affected.  >     1) Ton knows accelerometers better than his wife :-).  I hope not...  >     I ask you, Ton, and all readers: should we conclude that the force field in a non-inertial frame is different from zero? (I remind you Necip Berne's statement about the need for equilibrium, i.e. net force = 0).  >  > MY ANSWER: Probably not.  My answer: yes.  The force field is not zero and it needs to be added to the equations  of motion in a non-inertial reference frame.  Only then will the movement obey the  equation sum(Fm) = m*Am.  The force field becomes one of the forces on the left hand side.  >     Well, what's the meaning of Ton's statements then? Did he maintain that accelerometers can measure apparent-imaginary-fictitious (i.e. inertial) forces?  >  > MY ANSWER: No, I don't think so. Ton perfectly knows that an accelerometer is "simply a mass attached to a little force transducer" (or some equivalent device embedded in an integrated circuit).  My answer: yes.  An accelerometer measures exactly the force that is required to  keep a mass from moving in the local frame.  And that is exactly the same force  (after dividing by the accelerometer mass, and multiplying by the particle mass)  that must be added to the equation of motion for a particle that moves around  in the local frame.  > I wonder how should we describe the net external force and the weight of the small  > mass. Ton, should we call them imaginary or real forces?  You can call them what you want, it is not important.  However, when explaining this  is might be helpful to say that some of the force is "real" (gravity) and some of it  is caused by accelerating or rotating the frame to which the accelerometer is  attached.  > But only one wrote Einstein's equivalence principle correctly: Dr. Kris Kirtley.  >  > "..cannot be any 'interior' experiment."  Thanks for clarifying this.  This is indeed an important part of the principle.  >     Referring to Ton's example about weather forecasting, please let's not forget we  > need to know the angular velocity of the earth and the radius of rotation of the air  > particles to compute the value of the Coriolis and centrifugal forces thei are acted  > upon. Ton, I have a latter question for you: were these data measured in a "fixed" -  > inertial reference frame or in a non-inertial reference frame?  Good question.  These data were probably derived by observing the motion of the  stars relative to the earth, then assuming that the stars do not move so this  must reflect rotation of the earth.  So an inertial frame (the stars) was used.  However, if we had cloudy skies and could see no stars, we could have measured  these forces with a (very sensitive) accelerometer.  Ton van den Bogert  --  A.J. (Ton) van den Bogert, PhD  Department of Biomedical Engineering  Cleveland Clinic Foundation  9500 Euclid Avenue (ND-20)  Cleveland, OH 44195, USA  Phone/Fax: (216) 444-5566/9198
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