This post concerns a debate at the Midwest Graduate Students
Biomechanics Symposium on March 16. I apologize if I do not represent
the other party's arguments very well and invite his participation.
I also apologize for a longish post. However, I felt that the
debate was ended prematurely at the symposium and that further
discussion would be in order. It was suggested at the symposium that
debates such as these take place on the internet, which I agree to.
I welcome any feedback, particularily if I have made an error in my
thinking, which has been known to happen but which I do not believe
has occurred in this instance.

A statement was made that muscle forces are greater during the
concentric phase than during the eccentric phase. To prove this
vertical ground reaction forces were measured while the subject
completed an elbow curl holding a 25 pound weight. The ground reaction
forces showed greater deviations, both positive and negative, from
the weight of the individual plus twenty-five pounds during the
concentric phase than during the eccentric phase. This was given e
as proof that the muscle forces were larger during the concentric
phase. It is my assertion that this is not necessarily, or even
generally, true.

My Comments:
1) This test is not a good one for answering the question. What
the vertical ground reaction forces will show is the amount of
acceleration of the center of gravity of the body plus the weight.
The greater deviation in forces during the concentric phase
indicates that greater accleration occurred during the concentric
phase in this instance. However, by moving the arm slowly during
the concentric phase and dropping rapidly during the eccentric
phase I was able to produce greater deviations in force during
the eccentric phase. The point here is that this will not be
consistent and does not provide a good test for when muscle forces
will be greatest. To actually test to see when the muscle forces,
or more accurately, the muscle moments are greatest would require
collecting film or video data and would not be dependent on
force plate data.

2. (I apologize for not being able to write formulas very well with
the mail system I am using)
The moments at the elbow at any given point in time will follow


where M(muscle) is the moment caused by the muscles and other
assorted forces (ligaments, bone-on-bone contact, etc.) and M(weight)
is the moment caused by the weight of the arm plus the external
weight (and is given as a negative).
By taking the integral of both sides of the equation, we get


Since the starting and ending conditions are at omega=0,
Integral(I*alpha)=0 so the equation ends up as


The value of M(weight) is not dependent on whether the movement
is concentric or eccentric so it is clear that the sum of the
muscle moments over time will be the same regardless of whether
the movement is concentric or eccentric. While at some points
during the movement the moments may be greater during one phase
rather than the other, the total moments must come out the same.
In fact there are two situations where the muscle forces
throughout the movement would obviously be the same. If the
motion occurs at a constant angular velocity, then alpha=0. If
this is true, than at any time and at any angle


Since M(weight) is determined by the angular position than
M(muscle) will be the same at a given angular position whether
the forearm is moving up or down.
It may be argued that, in order to start and stop at either
end of the movement, the assumption that alpha=0 would not be
correct. If, however, the movement is a "mirror image", that is
accelerations are the same going up and down at any angle
(alpha=f(angle)) once again we have


which becomes




Once again, the moment at any given angle is a function of that
angle and not of the direction of movement.
Given that I have provided two examples of situations where
the muscle force during the concentric phase is not greater than
the muscle force during the eccentric phase I think that it is
clear that it is not always true that concentric muscle forces
are larger than eccentric muscle forces.
The only counterargument I can think of is that it may be
that the antagonist muscle activity (for example the triceps)
is greater during the eccentric phase. I am fairly certain,
though, that this was not the point that was being made in the

Thank you for your help,
Phil Fink
Ph.D. Student
Purdue University