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DRAFT
02 Jan 2017
What puts the spring in your
step?
02/01/2017
The journey of a thousand miles begins with one step. Lao Tzu
Abstract
Walking is an interaction between two closed structural systems, the walking organism and the
earth, Newton’s Laws of Motion govern their interaction. The first step of the organism requires
an outside force to initiate movement. The spring-mass model for walking will require an
external force to load the “spring”. This paper proposes that gravity can be used to load the
spring and that much of that springiness will be stored in the bones as potential energy.
Kewords: Walking, Spring mass model, Closed kinematic chains, internal force, external force
External forces are forces caused by external agent outside of the system. Internal forces
are forces exchanged by the objects in the system. To determine what part should be
considered external and internal, mechanical system should be clearly defined.
https://www.boundless.com/physics/textbooks/boundless-physics-textbook/linear-momentum-and-collisions-7/
conservation-of-momentum-69/internal-vs-external-forces-295-5222/
A biologic organism is, in Newtonian terms, a close system, a structurally independent
body that needs an external force to interact with another external body. Within the skin
of the organism, the internal organs, including the locomotor system, function as part of
that closed system and cannot interact with other external closed systems without the
intervention of an external force. To get a body to move in relation to another body, such
as the organism moving on the earth, requires the action of an external force. The
kinetic energy taken into a body can be manipulated, but what energy that comes out
cannot be more, or less, than what goes in. Manipulating the incoming energy may be a
function of the internal objects but the exiting energy will never be more than what is put
in. When a bullet shell explodes within a gun barrel, the force of recoil is equal to the
force at the end of the bullet, it is a manipulation of mass and velocity that make it seem
different. Just like with a bullet with its energy coming from the gunpowder, there is
stored internal energy in living bodies that originate in an outside force, the food we eat,
that manifests as tissue “prestress”, the intrinsic tension present in all tissues, but that
energy is limited and rapidly depleted. Its purpose is mostly to keep the internal engines
of life functioning and to prime the pump to facilitate the intake of outside forces.
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If you are floating in space, throwing a ball is problematical, you mat be able to
accelerate the ball away from you with your stored internal energy, but you would be
pushed back with equal and opposite force away from the ball. The usual force
generated to throw a ball while standing in place comes from the GRF with kinetic
energy from the GRF converting it to potential energy within the body, and sending it out
again as kinetic energy with the ball movement. The throwing system, the internal
mechanism of the body, is the vehicle that serves as a passthrough for the external
energy. It may take that energy and do various things with it, but in the end, the output
energy will equal the input energy. In our everyday world gravity is the main force to be
reckoned with. A diver coming off a diving board can twist, turn, and distort their body,
that is the internal forces working, but all the twists and turns cannot affect the pull of
gravity. The diver’s body will fall at the rate of 9.8m/sec/sec no matter how hard the
internal forces work.
The first step
A ball sitting on the floor cannot move itself, an additional external force must be applied. It is in
equilibrium, happily obeying Newton’s First Law of Motion, gravity pulling it in toward the center
of the earth and the normal forces from the floor blocking it and pushing out with equal force.
Standing in place with two feet on the ground, a living body cannot move unless they can
muster additional external force to push against the floor. From where? Muscles? As illustrated
in the diver analogy, muscles are an internal force, an additional external force is necessary.
Some energy is stored in muscles, (because of the stored energy they are never lax), and the
same can be said of ligaments and all collagenous tissues as collagen always has some
intrinsic tension within its fibers, and the bone and soft tissues are additionally preloaded with
the compression force of bodyweight. However, when standing, the collagenous tension is
there to hold you upright, merely tensing muscles further gets you no place. To move, gravity
and the GRF must be invoked either by raising it, getting up on the toes, or lowering it, gaining
potential energy by compressing inner springs. By a clever shifting of mass, (an internal
operation), the external acceleration forces of gravity can be recruited and increase load can be
placed over one limb, compressing the floor bit more with the mass times the acceleration
forces of gravity. The floor pushes back, compressing out internal springs with equal vigor
{Roberts 2011} and the first step ensues as the weight is off the loaded limb allowing the bony
spring to release its potential energy . Subsequent steps follow with a smoother version of a
duck waddle, shifting the center of mass to take advantage of the gravity created kickback from
the floor. This refined duck waddle version of walking is the application of the long accepted
spring mass model for running, {Blickhan 1989}{Geyer 2006}.
The “six determents of gait” and the “inverted pendulum” models for walking {Kuo 2007} do not
model the all important first step as there seems to be no recognition that an external force is
necessary to move a body at rest. The internal muscle forces main function seems to be to
reconfigure the body shape and tension the system so as to then marshal outside forces to act
on it. The outside force is either gravity or the energy stored in the internal body springs as
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potential energy, put there by the Krebs cycle which converts outside energy to potential internal
energy, as springs can only return energy put there by an external force. Whether recognized
or ignored, a first step always involved a weight shift and that weight shift will load the body’s
inner springs. The most efficient use of that stored energy is to assist or replace the inner body
springs to minimize the body energy requirements and lessen the metabolic costs. A “push off”,
using the internal reshaping of the body by shortening or lengthening muscles actively pushing
against the ground requires a higher metabolic energy expenditure than a ‘bounce off” when
the body just bounced from the ground using the potential energy garnered from the GRF and
the muscles were used merely to reconfigure the body and shift its center of mass. Even a
push off is an enlistment of external forces, gravity, as it requires lowering the body center of
mass (COM), which gains potential energy.
Spring-mass Model
As already noted, springs can only return energy previously loaded by an external force, they
cannot produce mechanical energy on their own. The spring mass model for running envisions
the body as a pogo stick like structure, bouncing along using an interchange of potential
gravitational energy fed into the body on from the ground reaction and converting it to kinetic
energy for body motion. Alexander and Vernon {Alexander 1975} estimate that in some
animals, 70% of the energy during landing can then be used for take off. Blickhan states that
“the animal’s musculoskeletal system can be considered as an actively-driven, nonlinear,
multicomponent spring-mass system” {Blickhan 1989}. Alexander and Vernon attribute most of
the energy storage to muscle stretch, but physical laws dictate that when parallel elastic
materials are loaded it is the stiffer material that absorbs the greatest amount of kinetic energy
and converts it to potential energy within the stiff structure [Fig. 1] {Collins 2003}. It is, therefore,
axiomatic that the bone is the main energy storing device of the compressed contact limb. Bone
can be as hard as aluminum silicate but much more resistant to fracture. It might be compared
to an ash ax handle, stiff but springy. (The biblical Samson used the resilient jawbone of a
recently killed donkey as an effective club to kill a thousand Philistines {Bible
NewInternationalVersion 1973}).
The stored potential energy in bone can be dissipated as heat and be wasted in producing
sweat, or it can be utilized my the locomotor system to produce movement. The stiff bone
springs can shorten very little and that small motion would need to efficiently convert to a large
movement in a complaint limb {Geyer 2006}. How that is done is well described by Cleland,
{Cleland 1867}, and Lombard, {Lombard 1903}, who discuss mechanisms in which small
movements in the “short” muscles, (muscles passing over only one joint), can produce large
limb movements. Cleland shows that the same or similar organization of bone muscle and
ligaments are in humans, horses, and birds and Lombard describes it in a frog. Both authors
assume the that the action of “long” muscles, (muscles that span more than one joint), function
continuously, isometrically, and as adjustable tension ligaments. (Although Cleland and
Lombard did not use the physics terminology that can be ascribed to the linkage mechanisms,
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possibly because it was still an emerging concept amongst engineers ,{Reuleaux 1876}) , they
were clearly defining the locomotor system as operating as closed kinematic chains (CKC)
(Levin et al 2017). Energy in bones would pass to the contiguous ligaments and, somewhat
analogous to the bullet-gun recoil, a small movement in the bone can materialize as increased
tension in a ligament that, in turn, operating through a CKC , manifests as larger movements in
the more compliant soft tissues. These long muscles apparently perform best if they remain
isotonic, or nearly so, {Alexander 1977} {Biewener 1998} {Roberts 2011} and muscles can act
as ligaments. {Cleland 1867} As the spring action is occurring at multiple levels {Roberts 2011} it
is not difficult to visualize a chain of events that would link the bony springs to the taut
ligamentous system and into the muscles and tendons. It then becomes the potential energy
stored in the bone that powers the bone-muscle-ligament complex.
With the spring-mass model {Geyer 2006} supplanting the inverted pendulum model for walking
and running and integrating it into what has been know for over a century {Cleland 1867}
{Lombard 1903} about how muscles function in a coordinated CKC, we can develop a model
that brings gravitational external forces into the body and distributes them to efficiently mobilize
a body at rest and energize it to perform work in the most energy efficient manner.
Walking
As discussed above, The initial step is a weight shift to one leg and a recoil from the potential
energy stored in the bones and elastic soft tissues. As soft issues including the interstitial fluids
are nonlinear, it may be difficult to know when they are ‘stiff’ or compliant and just how much
bounce you can get from them {Noel 2017}, but we do know that bones, with a Young’s modulus
of cortical bone measured ultrasonically and mechanically was 20.7 GPa (S.D. 1.9) and 18.6
GPa , {Rho 1993}, are stiff, elastic, and will spring back went loaded. Adhering to the energy
conservation concept, the muscles and ligaments would be most energy efficient if they remain
isometric and isotonic. Internal forces would reconfigure the body thrusting the free limb forward
and shifting the center of mass, and this could be efficiently accomplished by utilizing the CKC
mechanisms as described by Cleland and Lombard [Fig. 2] . Walking would be a controlled
movement emanating from the central core out {Gracovetsky 1988} and maximizing the use of
the the potential energy garnered from the elastic recoil generated by the GRF. Angular changes
at multiple joints would be part of a coordinated CKC that could be controlled by a single
muscle or ligament within the CKC.
Energy Storage
The accepted wisdom is that muscles are our motors. That may be true for internal body
movements, but external movements may have adopted a different strategy. Wolff’s Law [Wolff
1889] and Davis’ Law [ Davis 1867] tell us that anatomy is a force diagram, the stiffer and
denser tissues at re those subject to the most forces. It becomes obvious that the mid shaft of
bones like the femur and humerus take the most compression forces and the tough ligaments,
such as those enveloping the pelvic girdle, cruciate ligaments of the knee, the Y ligament of the
hip, are resisting the greatest tension. As stiff structures in parallel with more malleable and less
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springy structures store more energy, then they will become the main source of the kinetic
energy that drives the body. If the body is handling the kinetic energy through pantograph-like
closed kinematic chains, {Levin et al 2017} then small movements at the source may generate
large movements and great speed at the interface with the external world.The tough pelvic
ligaments and the small amount of movement at the sacroiliac joints may be the force that
powers the spinal engine {Gracovetsky 1988}.
Ground Reaction Force
A pogo stick would slide out from under if it did not strike the ground with a normal (right angle)
force. Forward movement on a pogo stick is a factor of mid air weight shift, not angular forces
generated at ground contact. Efficient walking should mimic pogo stick/spring mass movement.
Foot contact at other than normal would generate underfoot shear that would waste energy. To
conserve energy, foot strike should be normal to the ground with all forces in and out of the
earth, not with angular forces that would generate shear. “Push-off”, actively accelerating stored
potential energy so they energize the ground to push back, requires energy expenditure that
the body would need to restore, so efficient walking would use that technique sparingly. A simple
weight shift at heel strike on the forward foot would require minimal internal forces and be the
most energy efficient. An energy efficient spring mass model for walking would use the energy
interchange between the outside gravity forces and the body mass to do then most work and
save the inside energy forces to reconfigure the body to take as advantage of the potential
energy garnered from the interchange with earth. At heal strike of the forward foot, weight shift
and body reconfiguring would “liftoff”, (an internal reconfiguring ), rather than “push-off” the
trailing limb with an external force pushing against the limb [Fig. 3]. It would be the internal
spring of the trailing limb contracting while the springs in the forward limb are expanding. Energy
dissipating shock absorbing footwear would waste energy, so shoes should protect the sole of
the foot from injury, but transmit forces without dampening them.
Conclusion
The spring-mass model appears to be a workable model for walking and, by inference, other
activities that require interchange of forces between the living body and the external world.
Organisms should be recognized as closed structural systems and that the interaction between
closed structural systems are subject to Newtonian Laws of Motion. For a body at rest, an
external force is required and movement in a spring-mass model requires an external source of
energy, usually gravity, to keep it active. An enduring rule in biology is that an organism is
always attempting to minimize energy expenditures and any modeling using spring-mass
concepts must take this into account. Using the elastic energy created in bone by its interaction
with earth would maximize the efficiency of the system interactions and seem a logical avenue
to explore when modeling biological activity of any sort.This paper expands on the spring-mass
model proposed by Alexander {Alexander 1976}, by updating the physics to conform to
Newtonian Laws of Motion, and integrates CKC mechanics that are consistent with anatomical
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observations recognized over a century ago,{Cleland 1867} {Lombard 1903} and recently
updated, {Levin et al 2017}.
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Figure 1. Stiff spring and soft spring in parallel. The stiff spring takes the
greater load.
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"7
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smLevin 2008
Figure 3. Weight shift about to occur at heel strike. Note - weight is still on the rear
foot and Center of Mass is shifting toward the forward foot. At heel strike the trailing
foot is lifted.
Figure 2. Spring mass walking using energy storage of bone.
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02 Jan 2017
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