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pdleva23
01-18-2010, 12:51 PM
I enjoyed reading Ton van den Bogert's very informative and thought
provoking comment regarding the concept of moment arm in 3-D (Re:
Terminology Query, 8 January 2010). I discovered with surprise that a
"moment arm matrix" exists, which I knew nothing about.

Among other things, Ton wrote that each of the scalar components (Mx, My,
Mz) of the moment M of a force F can be used to compute a different "moment
arm":

- Moment arm about x = |Mx| / |F|
- Moment arm about y = |My| / |F|
- Moment arm about z = |Mz| / |F|

Thus, in 3-D, and with respect to a given 3-D coordinate system, a single
force F has three moment arms (although, of course, each of these moment
arms may have length zero).

I do not completely agree about Ton's definitions, and I believe that eight
different moment arms "of F" with respect to the origin and axes of a given
3-D coordinate system can be defined. I will define them in the following
sections, and in the terminology section I will also maintain that not all
of them should be called moment arms "of F".

Providing a clear geometrical interpretation of their mathematical
definitions is another major goal of this contribution. The human brain is a
very powerful geometrical analyzer of 3-D space. Thus, it is crucial to me
to find the geometrical interpretation of a definition whenever this is
possible. This is the best way to allow my brain to truly grasp the concept
and make it unforgettable, and I hope this is true for my readers as well.


THE REJECTIONS OF F

Let me lead you gradually to my conclusion, by eliciting an apparently minor
doubt. A superficial analysis of the above-mentioned definition is likely to
lead you, at least initially, to the misleading conclusion that there is a
moment arm for each Cartesian axis (I purposely used the word "lead" three
times in this paragraph).

Actually, there's a moment arm for each plane of the coordinate system
(that's why we have only one moment arm in 2-D). You can more easily
understand that by considering that:

- since Fx has no moment about x, Mx is produced by Fyz
- since Fy has no moment about y, My is produced by Fzx
- since Fz has no moment about z, Mz is produced by Fxy

where Mx, My, Mz, and Fx, Fy, Fz are the vector (not scalar) components of M
and F, and

- Fyz = F - Fx (= Fy + Fz)
- Fzx = F - Fy (= Fz + Fx)
- Fxy = F - Fz (= Fx + Fy)

This means that Fyz, Fzx, Fxy are the vector (not scalar) projections of F
on the three Cartesian planes yz, xz, xy. Notice that:

- Fx, Fy, Fz are also called the "projections of F along x,y,z"
- Fyz,Fzx,Fxy are also called the "rejections of F from x,y,z"


MOMENT ARMS ABOUT CARTESIAN AXES

Thus, to really understand the definition of moment arm in 3-D, we need to
consider the projections of F on the three Cartesian planes (i.e. its
rejections from the three Cartesian axes). This trick leads us to a simple
definition of the moment arms "of F" (actually of its rejections) about x,
y, and z.

Let Fr be the rejection of a force F from a given axis of a 3D coordinate
system. The moment arm of Fr about the same axis is defined as the
perpendicular distance of the line of action of Fr from the same axis.

For instance, the moment arm of Fyz about x is defined as the perpendicular
distance of the line of action of Fyz from x. Mathematically, this means
that:

- Moment arm about x = |Mx| / |Fyz|
- Moment arm about y = |My| / |Fzx|
- Moment arm about z = |Mz| / |Fxy|

(notice that I am not using the magnitude of F, as in Ton's definition).

Since Fr lies, by definition, on a Cartesian plane (e.g. Fyz = F - Fx lies
on plane yz), together with the rotation center O, this definition is as
easy to understand as that of the moment arm in 2-D.

Yet we are not satisfied. To truly understand it, we need to prove its
validity. For that, we need to thoroughly reduce the 3-D problem into three
2-D problems by also considering, together with the three components of M,
and the three rejections of F, also the three rejections of d from x, y, and
z (d is the directed distance from O to the point of application of F):

- dyz = d - dx
- dzx = d - dy
- dxy = d - dz

Since M = Mx + My + Mz and since Fx has no moment about x, Fy has no moment
about y, and Fz has no moment about z, the basic equation in 3-D:

- M = d x F (a cross product)

can be validly decomposed into three simpler equations:

- Mx = dyz x Fyz (a cross product in which |dx| = |Fx| = 0)
- My = dzx x Fzx (a cross product in which |dy| = |Fy| = 0)
- Mz = dxy x Fxy (a cross product in which |dz| = |Fz| = 0)

which by the definition of the cross product imply:

- |Mx| = |dyz| * sin(alpha) * |Fyz|
- |My| = |dzx| * sin(beta) * |Fzx|
- |Mz| = |dxy| * sin(gamma) * |Fxy|

where

- |dyz| * sin(alpha) is the moment arm of Fyz about x
- |dzx| * sin(beta) is the moment arm of Fzx about y
- |dxy| * sin(gamma) is the moment arm of Fxy about z

Isn't that nice? By using a simple and therefore intuitively appealing
geometrical approach, based on projections on Cartesian planes, we obtained
a proof of the validity of the definition enunciated above. Notice that d,
dyz, dxz, and dxy are not necessarily perpendicular to the respective forces
F, Fyz, Fzx, and Fxy.

We will discover in the next section that, surprisingly, there is an even
simpler and computationally more efficient way to compute these moment arms.

(By the way, I would call O the "rotation center"; you might also call it
"fulcrum", but only if the rotating object is a lever, which is not a
necessary condition for this definition to be valid)


MOMENT ARM ABOUT THE ORIGIN

For a given coordinate system, in my opinion there's a fourth moment arm
that you can compute and that has an intuitively appealing practical
meaning. This is simply defined as the perpendicular distance of the line of
action of F from O. This moment arm does not necessarily lie on a Cartesian
plane (xy, yz, or zx). It lies on the plane defined by d and F.

To compute its value, we do with the cross product d x F the same thing we
did with the three cross products operating on the rejections of d and F
(see previous section). Since M = d x F, then, by the definition of the
cross product,

- |M| = |d| * sin(theta) * |F|

where

- |d| * sin(theta) is the moment arm of F about O

You can compute this moment arm even more easily and efficiently by using an
orthogonal cross division (OCD; de Leva, 2008), to compute the "orthogonal
anticrossproduct" (antiCP) of the cross product d x F:

- d_orth = M /o F = (F x M) / (F * F)

where "/o" is the symbol for the OCD and F * F is a dot product. The moment
arm is the magnitude of the orthogonal antiCP of d x F:

- |d_orth| = |M /o F|

This is computationally more efficient because you need to use the square
root only once (to compute the magnitude of M /o F), rather than twice (to
compute separately the magnitudes of M and F).

Don't be scared by the expression "orthogonal antiCP", it refers to an
elementary and widely known concept, I just invented its name and described
a handy vectorial operation to compute it (the OCD). The orthogonal antiCP
of d x F is just the component of d orthogonal to F, which means that

- d_orth x F = d x F

I cannot show you pictures in this message, so please see the picture
showing the orthogonal antiCP in my paper (fig. 4 in de Leva, 2008,
http://dx.doi.org/doi:10.1016/j.jbiomech.2007.09.030). It took to me about
one month to draw it! You will immediately understand what I mean.


MOMENT ARM VECTORS

There's yet another doubt which is worth mentioning. Are we sure that the
moment arm is just a scalar? If you agree that it has a direction, then you
also accept that it can be a vector. We have seen that there are four scalar
moment arms about O for each coordinate system. The OCD is the most
efficient way to compute the corresponding vectors:

- Moment arm vector of F = d_orth = M /o F
- Moment arm vector of Fyz = dyz_orth = Mx /o Fyz
- Moment arm vector of Fzx = dzx_orth = My /o Fzx
- Moment arm vector of Fxy = dxy_orth = Mz /o Fxy

Each of these is also called the orthogonal antiCP of the respective cross
product:

- M = d x F
- Mx = dyz x Fyz (a cross product in which |dx| = |Fx| = 0)
- My = dzx x Fzx (a cross product in which |dy| = |Fy| = 0)
- Mz = dxy x Fxy (a cross product in which |dz| = |Fz| = 0)

Similarly, the most efficient way to compute the four scalar moment arms
about O is:

- (Scalar) moment arm of F = |M /o F|
- (Scalar) moment arm of Fyz = |Mx /o Fyz|
- (Scalar) moment arm of Fzx = |My /o Fzx|
- (Scalar) moment arm of Fxy = |Mz /o Fxy|

How do you like it? Unfortunately, I believe that the three components of
d_orth are not equal to the moment arms of Fyz, Fzx, and Fxy. We can't have
everything!


MOMENT ARM ABOUT ANY AXIS

The trick of computing the rejection of F makes it extremely easy to compute
the moment arm of any rejection F-Fw about any axis w, even if w does not
coincide with an axis of your coordinate system:

- Moment arm of (F-Fw) = Mw /o (F-Fw)
- (Scalar) moment arm of (F-Fw) = |Mw /o (F-Fw)|

where Mw is the projection of M on w, and F-Fw the rejection of F from w.
Even in this general case, the traditional method is computationally less
efficient, and can only be applied to compute the scalar moment arm:

- (Scalar) moment arm of (F-Fw) = |Mw| / |(F-Fw)|

Notice that this should not be called the moment arm "of F" about w, but the
moment arm "of F-Fw" about w (see next section).


TERMINOLOGY

There are four (scalar) moment arms for each force with respect to a given
coordinate system, and since each of these moment arms has a direction, for
each of them there is a moment arm vector. Since the origin of this
discussion was a posting by Bill Sellers with the subject "Terminology
Query" (7 January 2010), it is worth highlighting that these eight moment
arms can be most effectively referred to as:

- (Scalar) moment arm of F about O
- (Scalar) moment arm of Fyz about x
- (Scalar) moment arm of Fzx about y
- (Scalar) moment arm of Fxy about z
- Moment arm vector of F about O
- Moment arm vector of Fyz about x
- Moment arm vector of Fzx about y
- Moment arm vector of Fxy about z

Or, equivalently (but only if you specify that the point of applications of
Fyz, Fzx, and Fxy, are dyz, dzx, and dxy, respectively)

- (Scalar) moment arm of F about O
- (Scalar) moment arm of Fyz about O
- (Scalar) moment arm of Fzx about O
- (Scalar) moment arm of Fxy about O
- Moment arm vector of F about O
- Moment arm vector of Fyz about O
- Moment arm vector of Fzx about O
- Moment arm vector of Fxy about O

Notice that the projection of F on the three Cartesian planes does not
entail necessarily that their points of applications are also projected on
the same planes. In other words, Fyz, Fzx, and Fxy might quite legitimately
have the same point of application of F (i.e. d), unless you specify
otherwise.

Unfortunately, although we can say:

- moment of F about x (Mx)
- moment of F about y (My)
- moment of F about z (Mz)

in my opinion it would be improper to say:

- moment arm of F about x [improper terminology]
- moment arm of F about y [improper terminology]
- moment arm of F about z [improper terminology]

Although these is intuitively appealing terminology, these expressions are
not exactly synonyms to above mentioned moment arms of Fyz, Fzx, and Fxy.
Let's not forget the reason why we needed the rejections of F (Fyz, Fzx,
Fxy):

- since Fx has no moment about x, Mx is produced by Fyz
- since Fy has no moment about y, My is produced by Fzx
- since Fz has no moment about z, Mz is produced by Fxy

Yes, Fx, Fy and Fz have no moment about the respective axes, but they do
have non-null moment arms (the perpendicular distances of their line of
action from the respective axes are not null). These moment arms are
meaningless, but unfortunately they exist, and this makes the
above-mentioned improper terminology meaningless, unless you explicitly give
it a different operational definition whenever you use it (e.g.: "I will
herein use the expression 'moment arm of F about x' to refer to the 'moment
arm of F-Fx about x'").


CONCLUSION

There are four (scalar) moment arms for each force with respect to a given
coordinate system, and since each of these moment arms unquestionably has a
direction, for each of them there is a moment arm vector. These eight moment
arms can be most effectively referred to as:

- (Scalar) moment arm of F about O
- (Scalar) moment arm of Fyz about x
- (Scalar) moment arm of Fzx about y
- (Scalar) moment arm of Fxy about z
- Moment arm vector of F about O
- Moment arm vector of Fyz about x
- Moment arm vector of Fzx about y
- Moment arm vector of Fxy about z

where Fyz, Fzx, Fxy are the projections of F on the Cartesian planes yz, zx,
xy, also called the rejections of F from the Cartesian axes x, y, z:

- Fyz = F - Fx
- Fzx = F - Fy
- Fxy = F - Fz

The most efficient way to compute all of these quantities is by means of the
orthogonal cross division (OCD; symbol: "/o"). For instance, the moment arm
vector of F about O is

- M /o F,

and its magnitude is the scalar moment arm of F about O:

- |M /o F|.

In the general case, by using an OCD you can compute the moment arm of F-Fw
about any axis w, even if w does not coincide with an axis of your
coordinate system:

- Moment arm of (F-Fw) = Mw /o (F-Fw)
- (Scalar) moment arm of (F-Fw) = |Mw /o (F-Fw)|

where Mw is the projection of M on w, and F-Fw the rejection of F from w.
The traditional method is computationally less efficient and can only be
used to compute the scalar moment arm:

- (Scalar) moment arm of (F-Fw) = |Mw| / |(F-Fw)|

Both the traditional method and the method based on the OCD have a clear
geometrical interpretation, since they both require the computation of the
rejections of F from x, y, z, or w. For instance, the moment arm of Fyz
about x can be geometrically defined as the perpendicular distance from x to
the line of action of Fyz, and since Fyz is also perpendicular to x, this is
not difficult to understand.

The concept of anticrossproduct (antiCP) is a useful tool for the
geometrical interpretation of several widely used formulas (see
http://dx.doi.org/doi:10.1016/j.jbiomech.2007.09.030), including the
definition of moment arms in 3-D. For each cross product such as d x F,
there are infinitely many antiCPs (similarly, for each derivative, there are
infinite antiderivatives). They are simply defined as the infinitely many
vectors xi which meet the condition xi x F = d x F. Obviously, they include
d. They are infinitely many, but only one of them is orthogonal to F.

Indeed, an OCD returns the "orthogonal antiCP" of the corresponding cross
product. For instance M /o F returns the orthogonal antiCP of d x F, which
may be also defined as the component of d orthogonal to F. Thus, the four
moment arm vectors listed above can be also called the "orthogonal antiCPs"
of the corresponding cross products.

By the way, perhaps Ton or someone else might be willing to prove that my
definition of the moment arm about a given Cartesian axis is equivalent to
the "delLen/delTheta" formula described by Bill Sellers in its original
posting about this topic (Terminology Query, 7 January 2010).


REFERENCE

de Leva P., 2008. Anticrossproducts and cross divisions. Journal of
Biomechanics, 8, 1790-1800
(http://dx.doi.org/doi:10.1016/j.jbiomech.2007.09.030)


With my kindest regards,

Paolo de Leva
Laboratory of Locomotor Apparatus Bioengineering
Department of Human Movement and Sport Sciences
University of Rome, Foro Italico
Piazza Lauro de Bosis, 6
00135 Rome - Italy
http://www.lablab.eu