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Re: More and more angles and powers

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  • Re: More and more angles and powers

    Dear Richard and all:

    I agree with what you wrote and want to add a simple 'geometric' explanation
    of what is happening when power is computed in the ICS axes. To make the
    understanding easier I replaced your vector a with F (force) and vector b
    with V(velocity).

    In the JCS system, the axes i and k are in the plane that is orthogonal to
    axis j (a floating axis). Let's introduce in this plane an axis l that is
    orthogonal to axis i and consider an orthogonal system of coordinates ijl.
    In this reference system, power FV can be represented as a sum of three
    terms associated with the individual force-velocity projections on the
    coordinate axes:

    FV=fi*vi+fj*vj+fl*vl

    Because power is invariant [its magnitude does not depend on the selected
    (inertial) system of coordinates] and the first two terms in the above
    equation are equally valid for the JCS system, the following equality is
    also valid:
    fl*vl= fk*vk+(fk*vi+fi*vk) (i.k)

    In this equation: fl*vl is a power term representing the power associated
    with the force and velocity components along the orthogonal axis l; fk*vk
    is a similar term representing the power associated with the non-orthogonal
    axis k, (fk*vi+fi*vk) (i.k) is the difference (DELTA) between the above
    power values. Because (i.k.) is simply a cosine of angle A formed by the
    axes i and k, the DELTA = 0 when k is along axis l (i.e. the system ijk is
    orthogonal) and DELTA is not equal to zero in all other cases. DELTA=1 when
    axes i and k are along the same direction. In the latter case the
    singularity occurs.



    Sincerely,



    Vladimir Zatsiorsky



    ----- Original Message -----
    From: "Richard Baker"
    To:
    Sent: Wednesday, January 28, 2004 5:49 PM
    Subject: Re: [BIOMCH-L] More angles and powers


    > Dear Jonas and all,
    >
    > Ton and I have had a brief exchange of notes on a sub-topic of the current
    > discussion - that of whether the decomposition of the dot product a.b
    > =ax.bx+ay.by+az.bz works in non-orthogonal axis system. I resorted to an
    > old textbook on mathematical physics which said "no" with some fairly
    heady
    > math to explain. Ton's come up with the same answer but in a much more
    > insightful manner:
    >
    > Let a and b be vectors, and i,j,k be unit vectors along the coordinate
    axes
    > which may be non-orthogonal. Let ai, aj, ak be the scalar components
    along
    > each axis. Then:
    >
    > a = ai.i + aj.j + ak.k
    > b = bi.i + bj.j + bk.k
    >
    > Hence:
    >
    > (a.b) = ai.bi.(i.i) + ai.bj.(i.j) + ai.bk.(i.k) +
    > aj.bi.(j.i) + aj.bj.(j.j) + aj.bk.(k.k) +
    > ak.bi.(k.i) + ak.bj.(k.j) + ak.bk.(k.k)
    >
    > = ai.bi + aj.bj + ak.bk +
    > (aj.bi + bj.ai).(i.j) +
    > (ak.bi + bk.ai).(i.k) +
    > (ak.bj + bk.aj).(j.k)
    >
    > In the JCS, axes 1 and 2 are orthogonal, and axis 2 and 3 are orthogonal,
    > so we lose the cross terms with (i.j) and (j.k). The (i.k) term
    > remains :-(. So it seems that you need to add this term if you wanted
    > to compute total power from JCS angular velocities and JCS moment
    > components.
    >
    > We are both agreed that vector relationships must hold whatever the
    > co-ordinate system used (whether orthogonal or not) but the way in which
    > these are calculated from the basic components will depend on the
    > characteristics of the co-ordinate system.
    >
    > Richard
    >
    >
    > Richard Baker
    >
    > Gait Analysis Service Manager, Royal Children's Hospital
    > Flemington Road, Parkville, Victoria 3052
    > Tel: +613 9345 5354, Fax +613 9345 5447
    >
    > Adjunct Associate Professor, Physiotherapy, La Trobe University
    > Honorary Senior Fellow, Mecahnical and Manufacturing Engineering,
    Melbourne
    > University
    >
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    >

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