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Re: Moments about C of R?

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  • Re: Moments about C of R?

    Dear Biomch-L readers,

    The posting on joint centers of rotation by Ian Stokes is interesting;
    a (literal) shift of perspective from the center of rotation can be useful.
    I tend to disagree however, with his final conclusion that "...neither
    biomechanical theory, nor practical considerations support it (the use
    of joint centers as reference points)". If you do not make assumptions
    about the line of action of the joint force (JF), the JF must be described
    by 3 variables (2D) instead of 2: two components of force, and one for
    the line of action. This introduces one extra unknown variable into the moment
    equilibrium equation, and typically there are too many unknowns already.

    So you must make an assumption about the line of action of the JF. But there
    is only one thing a priori known about this: the JF goes through the
    instantaneous center of rotation (ICR).

    This can be proved using the principle of virtual work. A joint is defined
    as a 'kinematic' connection, i.e. the force associated with this connection
    generates or absorbs no power at any time. Picture one body (bone) as stationary
    while the other is moving. All points on the line of action of the joint force
    must have velocities perpendicular to this force (power is the dot product
    of force and velocity). In a moving rigid body, every line on which all
    velocities have the same direction *must* pass through the ICR. Please take
    a few seconds to verify this statement...
    So, the joint force also passes through the ICR. Incidentally, this also
    proves that the ICR of the knee joint during the swing phase coincides with
    the intersection of the cruciate ligaments. The joint force is in that case
    the resultant of ligament forces only.

    Note that this only applies to true kinematic connections, without frictional
    losses or energy storage in elastic cartilage or joint ligaments. Neglecting
    these small amounts of energy is probably allowed. Also note that in this
    definition, 'joint force' is taken to mean the total 'constraint reaction
    force' in mechanical terms, sometimes called 'net joint force'. If you
    only want the contact force, without ligaments, the ligament forces become
    additional unknowns in the equilibrium equations and that is not what you
    want.

    A joint, defined as a kinematic connection between two bodies, is more than
    just the bone-to-bone contact surfaces. It also includes the structures
    that guide the movement without exchanging energy with the system. I.e.
    ligaments that can be considered inextensible for the purpose of dynamic
    analysis.

    Remains the problem that the ICR has (in general) no fixed position on either
    bone, and that the ICR is not easily determined during actual movements.
    That is exactly why the ICR is taken as the reference point in the moment
    equation. That way you do not have to know it! Of course, this implies that
    all other moments must alse be calculated about the ICR. For muscular
    forces this is no problem: the moment arm with respect to the ICR is the
    partial derivative of origin-insertion length with respect to the joint
    angle (also to be proved by the principle of virtual work). Using this
    definition, moment arms of muscles are easily determined from
    cadaver measurements or a rigid-body model incorporating the line of action.
    Only for calculation of external (ground reaction force) moments is an
    estimated location of the ICR required. Hopefully, moment arms of ground
    reaction forces are large enough (or the moments small enough) to be
    insensitive to errors in the ICR.

    So, my opinion is that moments should be calculated about the ICR. I would
    like to hear Ian's reply, or other opinions.

    -- Ton van den Bogert
    University of Utrecht, Netherlands.

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