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Summary:Modeling Muscle Stiffness in Athletes

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  • Summary:Modeling Muscle Stiffness in Athletes

    Dear Group:
    Last summer I attempted to post this summary. In spite of my efforts to
    send it as a text file, somehow it became an encoded file which only a few
    clever individual were able to decode. I have been asked to try again, so
    this time I hope it works.

    Clifford Larkins


    Dear Group:

    A few weeks ago I posted a request for help related to a project I have
    been working on for some time. I would first like to thank those who
    provided comments, criticisms, suggestions, and references. While some
    of the respondents specifically addressed the question I posed (related to
    modeling muscle stiffness in athletes), others commented on the limitations
    of using a simple Spring-Mass System to model complex athletic
    In order to stimulate an exchange of ideas, I replied to each respondent
    individually first, clarifiying my objective, and then commenting on their
    replies. I sometimes then raised further questions about the Spring-Mass
    In order to stimulate a discussion among the entire group, I have been
    encouraged by some of the respondents to post as many of the dialogues as
    possible‹not just a summary.

    The original message stated:

    Dear Group:
    I have developed a mathematical model which uses a simple spring mass
    system to model the long jump and the triple jump. I need help in
    determining spring stiffness values (k) which will realistically represent a
    jumper's muscle stiffness. Keeping all other input variables fixed, I varied
    the spring stiffness using a range of values from 500,000N/m to
    1,120,000N/m. This range was taken from McMahon and Green, 1979.
    These values revealed decreasing jump distances as the muscle stiffness
    increased in the range 900,000N/m to 1,200,000N/m.

    1) Are the k values I selected realistic for world class long
    and triple jumpers?

    2) Are the results reasonable: decreasing distance with increasing
    stiffness? I would have expected the opposite to occur.

    3) Would increased muscle stiffness correlate highly with
    increased fitness and training? In other words, as my athletes
    become more fit, should I input larger k values into my model?

    Answers to these questions and any suggestions will be greatly appreciated.

    Clifford Larkins, Ph.D.

    Clarification of objective:
    The objective of this project is to provide coaches with an interactive
    program which will allow them to manipulate various aspects of long jump
    and triple jump technique systematically and to see immediately the results
    of their coaching decisions. I also hope to use the program in a 500 level
    sports biomechanics course to teach basic concepts of jumping, physics,
    calculus, computer programming, muscle mechanics, and mathematical

    References: Two Proceedings Articles describe the model:
    Development and Use: Clifford Larkins and Manfred Vieten
    Theoretical Basis: Manfred Vieten and Clifford Larkins
    The proceeding for the XIth International Symposium of International
    Society of Biomechanics (ISBS) Annual Meeting on June 23-26, 1993 at
    the University of Massachusetts.

    From: Jacek Cholewicki
    Subject: Re: Help:Modeling Muscle Stiffness/Athletes

    Muscle stiffness is roughly proportional to its force and inversely
    proportional to its length. Bergmark came up with the following
    relationship: k=q(F/L), where q is about 30. When I did a quick
    calculation, assuming the maximum quadriceps force of 10,000N and its
    length of 30 cm, the stiffness k was about 1,000,000 N/m. You can find
    more info and references in: Cholewicki, J. and McGill, S.M. (1996).
    Relationship between muscle force and stiffness in the whole mammalian
    muscle: a simulation study. J. Biomech. Eng. 117:339-342.

    Jacek Cholewicki, Ph.D. Tel. (203)785-2812
    Assistant Professor Fax (203)785-7069
    Biomechanics Research Laboratory e-mail:
    Dept. of Orthopaedics & Rehabilitation
    Yale University School of Medicine
    P.O. Box 208071
    New Haven, CT 06520-8071

    Response From: Larkins
    Subject: Re: Help:Modeling Muscle Stiffness/Athletes

    Dear Dr. Cholewicki

    I can certainly see some practical application to Bergmark's formula. If
    an accurate anthropometric measurement procedure can be devised to
    measure the length of the hamstrings from origin to insertion, one could
    plug in individual k values for each individual athlete.


    Clifford Larkins, Ph.D.

    From: Edwin DeMont
    To: Larkins
    Subject: Re: Help:Modeling Muscle Stiffness/Athletes


    Have you seen the following paper:

    Alexander, R.McN. (1990). Optimum take-off techniques for long and high
    jumpers. Philosophical Transactions of the Royal Society. B329: 3-10.

    Alexander developed a model that sounds very much like


    Edwin DeMont, Ph.D.
    Biology Department
    P.O. Box 5000
    St. Francis Xavier University
    Antigonish, Nova Scotia
    Canada B2G 2W5


    Response From: Larkins
    Subject: Re: Help:Modeling Muscle Stiffness/Athletes

    Dear Dr. DeMont

    The essential difference is that Alexander's model is a researcher's model
    developed at such a high level of complexity that only a researcher with
    knowledge of muscle mechanics could use it. My model is a coach's model
    intended to be used by knowledgeable coaches ‹knowledgeable, that is, in
    the sense that they must understand basic concepts of mechanics as it would
    be taught in an undergraduate course in sports biomechanics. Therefore,
    the input parameters necessary for the S-M model were devised from the
    point of view of a coach.
    Of course, an important question is whether a S-M model is too simple to
    represent human jumping. With the possible limitations of this model in
    mind, Dr. Manfred Vieten and I refined the model so that an important input
    parameter would be an individual athlete's GRFs. If it is possible for
    coaches to collect GRFs on their athletes, as we suggested in the ISBS
    proceedings paper, they could use our refined model, the Advance S-M
    model. I believe that this is possible because I have taught physical
    education students to collect and interpret GRF data.


    Clifford Larkins, Ph.D.

    From: Mike Harwood
    Subject: Re: Help:Modeling Muscle Stiffness/Athletes

    Dear Dr Larkins

    I was very interested to read your message. I am also working on this
    type of model, but in my case I am looking at gymnastic activities,
    particularly vaulting.

    I don't have any figures with me but I can make a few comments which
    might be of interest:

    Question 1: "Are your k values realistic...?"
    The only figures I have seen for long jumping were reported by Blickhan
    at ISB congress in 1995. His model had a variable stiffness if I remember

    Question 2: "Are the results reasonable...?"
    One reason for a decreasing distance with increasing stiffness might be
    due to the angle of the velocity vector of the mass centre at takeoff.
    For stiffer springs I imagine your model leaves the ground sooner (i.e.
    at a steeper angle) if you keep all of the other variables the same.

    Question 3: "Would muscle stiffness correlate with fitness..?"
    I think you need to remember that your model has an overall stiffness not
    a single muscle's stiffness and also that the overall stiffness will to
    some extent be under voluntary control (e.g. McMahon, Valiant and
    Frederick's paper on Groucho Running, and more recently the paper by
    Farley and a coworker in J App Physiol (I think) Jan or Feb 1997).
    However I
    would think it reasonable to assume that suitable (strength) training of the
    appropriate muscles (e.g. hip, knee and ankle extensors) might affect
    the athlete's maximum stiffness for a given amount of knee flexion (for

    I hope this is of use. I am going to be away from my email link for
    nearly 2 weeks but if I can be of any more use please get back to me (if
    you need full references for example- sorry I don't have them to hand
    now). Also I would be very interested in hearing what others have to say.

    Best wishes and have a happy Easter


    Mike Harwood Voice: 01234 793353
    De Montfort University Bedford Fax.: 01234 350833
    United Kingdom email:

    Response From: Larkins
    Subject: Re: Help:Modeling Muscle Stiffness/Athletes

    Dear Mike,
    Here are some questions your reply stimulated.

    Question 1: Are there Proceedings for the 1995 ISB Congress? I didn't
    know Blickhan had written anything about the long jump.

    Question 2: Here you make a particularly interesting point. I will use the
    S-M model to check this out. It generates the question: If an increasing
    stiffness causes an increase in theta takeoff, then does it follow that the
    height of the jump will increase and the distance of the jump decrease? This
    appears to be what you are saying and what I am seeing in my data.
    Further, it drives me to ask myself: What does this tell us about S-M
    systems? Can this be generalized to humans?

    Question 3: If our conclusion in Question 2 is true, then there would be
    no advantage to increasing the muscle stiffness of a horizontal jumper. Is
    there (I must ask my self) a relationship between spring stiffness, muscle
    stiffness in humans, and the type of muscle strength and power I would
    train my athletes to achieve ?

    If you don't mind me posting your reply, I will paste together some
    kind of discussion summary. Maybe our thoughts will stimulate some
    answers from the group.
    Thanks again: your ideas are particularly stimulating. Feel free to contact
    me if you have any more thoughts, solutions, comments, or suggestions.

    Clifford Larkins, Ph.D.


    Subject: Re: Help:Modeling Muscle Stiffness/Athletes

    Dr. Larkins,

    Regarding your questions about stiffness and modeling, I think that you
    find a simple spring-mass model is inappropriate for modeling a long jump.
    One of the basic principles of a spring-mass system is that no energy can
    be added or lost from the system. Thus, it seems inconsistent with the
    nature of the long jump where the last step provides a large impulse to the
    center of mass.

    As for stiffness, the values reported in McMahon and Green (1979) are
    vertical stiffness values and not leg stiffness values. McMahon and Cheng
    (1990) discuss the difference between the two. As far as I know, there are
    leg stiffness data on only seven human runners that have ever been
    published. The leg stiffness data can be found in Farley and Gonzalez
    (1996) and He, Kram, and McMahon (1991).

    Best of luck,
    dan ferris

    Daniel P. Ferris
    UC Berkeley Locomotion Laboratory
    3060 Valley Life Sciences Building
    University of California
    Berkeley, CA 94720-3140

    Tel (510) 642-8662
    Fax (510) 643-6264
    Response From: Larkins
    Subject: Re: Help:Modeling Muscle Stiffness/Athletes

    Dear Daniel,

    You are correct to inform me that "one of the basic principles of a spring-
    mass system is that no energy can be added or lost from the system" and
    that "...the last step provides a large impulse to the center of mass."
    However, a long jumper's primary objective during the last step is to
    conserve as much of the kinetic energy developed during the approach as
    possible. In this way the jumper attempts to mimic a S-M system.
    Actually, the "large impulse" applied during the second half of the support
    phase is the jumper's attempt to conserve the horizontal velocity component
    and generate the necessary vertical velocity component. Because the jumper
    is not a simple S-M system there will indeed be a change in velocity from
    touchdown to takeoff ‹usually a loss. Given data from the numerous
    studies that have shown this, it is possible to factor a percent loss in the


    Clifford Larkins, Ph.D.

    From: Claire Farley
    Subject: leg stiffness

    Dear Dr. Larkins,

    Dan Ferris is a graduate student in my laboratory and he told me about the
    correspondence between him and you regarding the use of a spring-mass
    model for modeling a long jump. I agree with Dan about the concern that the
    model really only applies in situations where there is no net loss or gain of
    energy during the stance phase. However, I did read your note that a long
    jump is closer to fitting this criterion than I previously thought. One
    important thing to realize is that leg stiffness is not an invariant property of
    the musculoskeletal system. Studies on hopping in place show that leg
    stiffness can be adjusted to accommodate changes in hopping frequency,
    hopping height, or surface stiffness. Similarly, during forward running, leg
    stiffness is adjusted to accommodate changes in stride frequency during
    running at a given speed. The adjustability of leg stiffness depending on
    task makes it necessary to actually measure leg stiffness during the activity
    of interest (i.e., in your case, during a long jump). Our data are showing
    that leg stiffness can be changed by more than 3-fold depending on task and
    that is a big range. Some papers you might find helpful:

    Farley, C. T., R. Blickhan, J. Saito and C. R. Taylor. Hopping frequency
    in humans: a test of how springs set stride frequency in bouncing gaits.
    Journal of Applied Physiology 71(6): 2127-2132, 1991.

    Farley, C. T., J. Glasheen and T. A. McMahon. Running springs: speed
    animal size. Journal of Experimental Biology 185: 71-87, 1993.

    Farley, C. T. and O. Gonzalez. Leg stiffness and stride frequency in
    running. Journal of Biomechanics 29: 181-186, 1996.

    Ferris, D.P. and C.T. Farley. Leg stiffness adjustment to accommodate
    changes in surface stiffness during human hopping. Journal of Applied
    Physiology 82(1): 15-22, 1997.

    Best wishes,

    Claire T. Farley, Ph.D.
    Locomotion Laboratory
    3060 Valley Life Sciences Building
    University of California
    Berkeley, CA 94720-3140.

    Please note new postal address
    and E-mail address.
    ************************************************** **

    From: "Paolo de Leva - Sport Biomechanics, Rome"

    Subject: R: Help:Modeling Muscle Stiffness/Athletes

    Dear Clifford,

    In my opinion your model is too simple to behave as the human
    body. This has at least two consequences:
    a) Your model is only capable of eliciting ground reaction f
    orces oriented toward its CM. The human body also uses force
    components that
    are normal to the "foot-CM" direction. Although these forces change
    body angular momentum, at the same time change the take-off
    velocity, and particularly the take-off angle (direction of take-off
    velocity vector).
    Using these forces and other capabilities that your model doesn't
    have (and that make muscles very different from simple springs), an
    athlete can achieve an optimal take-off angle, which
    your model won't necessarily achieve. It seems plausible to me that
    your model will achieve an optimal take-off angle only with a given
    b) The optimal stiffness for your model has little to do with
    the optimal stiffness for a human muscle or muscle-bone system
    (provided that you can use a spring to model the behaviour of a
    system controlled by muscles, which is quite questionable, in my
    opinion). Thereforethe human muscle (or muscle-bone system)
    range is not necessarily good for your model.
    You might have better results using stiffness values OUTSIDE THE


    Even if your model were sufficiently similar to a human body, there
    would remain another aspect that you should consider.
    The stiffness of your spring has different effects depending
    on the landing angle (orientation of the "foot-CM" line), and on the
    horizontal velocity at landing.
    I guess there must be a different optimal landing angle for each
    stiffness value, and for each landing velocity.
    Therefore it seems to me that you should not "keep all the
    other input variables fixed". Similarly, but for much more complex
    (see CONCEPT 1), I bet that the landing angle is not the same for all
    the athletes. It might depend on landing velocity, training,
    anthropometrical parameters, etc.

    With regards,

    ________ _________ ___________~___ ________
    / ~ ~ ~ ~ \
    / \
    | Paolo de Leva ~ \ Tel.+ FAX: (39-6) 575.40.81 |
    | Istituto Superiore di Educazione Fisica > other FAX: (39-6) 361.30.65 |
    | Biomechanics Lab / |
    | Via di Villa Pepoli, 4 < INTERNET e-mail address: |
    | 00153 ROME - ITALY \ Pa.deLeva@Agora.STM.IT |
    |________ ____________~________~_______ ____________~_____
    ~ ~ Chal~enging entropy
    Response From: Larkins
    Subject: Re: Help:Modeling Muscle Stiffness/Athletes

    Dear Paolo,

    Response to Concept 1a -- Angular Momentum and S-M Models
    I agree that an important question is whether a S-M model is too simple
    to represent human jumping. Two of the four replies to my request for help
    insist that it is, but from what I can see these opinions are not based on
    evidence from testing a model. You are correct to inform me that a simple
    S-M model can not represent human performance in its most complex
    forms. However, my primary objective is to predict jump distance and for
    this purpose a simple spring mass model should be adequate. In fact, for
    centuries the "Ballistic Approach" (which is far simpler than S-M Models)
    has been used for this purpose in military, aeronautical, as well as athletic
    situations. The angular momentum developed by the eccentric forces
    generated during the final support phase causes the athlete to rotate while in
    flight‹usually forward. The angular momentum required to make a
    successful jump depends on the technique that the jumper prefers to use
    during his flight (see Ramey Dec. 73 "Significance of Angular Momentum
    in Long Jumping" Research Quarterly Vol. 44.). Therefore, if flight
    technique is not important in order to describe a successful jump (and it is
    not for my model), a S-M model should be adequate. The advantage of the
    S-M Model over the "Ballistic Approach" is that a S-M Model can be used
    to provide some hints (though very simple) about what occurred during the
    final support phase to generate the takeoff velocities. It also allows the user
    to manipulate approach variables and to see their affect on the distance

    Response to Concept 1a -- Optimum Takeoff Angles and S-M Models
    You indicate that because the S-M Model can not simulate the torques
    generated by human jumpers and also lacks "other capabilities" it will be
    unable to achieve optimum takeoff angle the way humans can. I believe you
    are over emphasizing the importance of achieving an optimum takeoff angle
    in the long jump. The long jump is not a target sport like archery or free
    throw shooting in basketball; therefore, long jumpers do not attempt to
    achieve an optimum angle of takeoff. Instead, what they do attempt to
    achieve is an optimum Vx and Vy coupling at takeoff which will yield
    maximum distance (which thereby generates a corresponding takeoff angle).
    Given that I can input realistic approach velocities and the model will return
    realistic takeoff velocity vectors, at this time I believe that I can achieve my
    stated objectives with this model.

    Response to Concept 1b -- S-M stiffness is not representative of Muscle
    From my preliminary tests, I believe you are right about this. I will test
    a wide range of spring stiffness in order to verify this observation. If you
    are right and the useful spring stiffness turns out to be outside human range
    of muscle stiffness, I don't think this will be a serious problem, given my

    Response to Concept 2 -- The Effects of spring stiffness on selected
    I also believe as you say, "there is a different optimal landing angle for
    each stiffness value, and for each landing velocity" and that "the [optimum]
    landing angle is not the same for all the athletes." Here you have stated a
    number of important hypotheses. It is the purpose of my model to test each
    of them.
    I deeply appreciate your reply because it impelled me to re-examine the
    mechanics of S-M systems as well as the mechanics of the long jump. Your
    reply also forced me to reexamine my objectives. Feel free to contact me
    with any further comments or suggestions.


    Clifford Larkins, Ph.D.