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

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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

movements.

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

System.

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.

Questions:

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.

E-mail: eng_harris@online.emich.edu

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

modeling.

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.

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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 jacek.cholewicki@yale.edu

Yale University School of Medicine

P.O. Box 208071

New Haven, CT 06520-8071

USA

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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.

Sincerely,

Clifford Larkins, Ph.D.

**************************************************

From: Edwin DeMont

To: Larkins

Subject: Re: Help:Modeling Muscle Stiffness/Athletes

Hello,

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

yours.

Cheers

Edwin DeMont, Ph.D.

Biology Department

P.O. Box 5000

St. Francis Xavier University

Antigonish, Nova Scotia

Canada B2G 2W5

email: edemont@stfx.ca

WWW: http://juliet.stfx.ca/~edemont/biomechanics-lab.html

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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.

Sincerely,

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

correctly.

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

example).

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.

---------------------------------------------------------------------------

Mike Harwood Voice: 01234 793353

De Montfort University Bedford Fax.: 01234 350833

United Kingdom email:

mharwood@dmu.ac.uk

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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.

Sincerely,

Clifford Larkins, Ph.D.

**************************************************

From: dferris@uclink2.berkeley.edu

Subject: Re: Help:Modeling Muscle Stiffness/Athletes

Dr. Larkins,

Regarding your questions about stiffness and modeling, I think that you

may

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

dferris@uclink2.berkeley.edu

http://garnet.berkeley.edu/~hbbiomxl/dferris/

Tel (510) 642-8662

Fax (510) 643-6264

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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

input.

Sincerely,

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

and

animal size. Journal of Experimental Biology 185: 71-87, 1993.

Farley, C. T. and O. Gonzalez. Leg stiffness and stride frequency in

human

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.

cfarley@socrates.berkeley.edu

http://socrates.berkeley.edu/~hbbiomxl/

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,

CONCEPT 1

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

the

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

stiffness.

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

living

system controlled by muscles, which is quite questionable, in my

opinion). Thereforethe human muscle (or muscle-bone system)

stiffness

range is not necessarily good for your model.

You might have better results using stiffness values OUTSIDE THE

HUMAN RANGE!

CONCEPT 2

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

reasons

(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

------------------------------------------------------------------------------------------

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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

jumped.

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

Stiffness

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

objective.

Response to Concept 2 -- The Effects of spring stiffness on selected

variables

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.

Sincerely,

Clifford Larkins, Ph.D.

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