Download The Biomechanics Behind Kicking a Soccer Ball

VANIER COLLEGE
The Biomechanics Behind
Kicking a Soccer Ball
The Energy Demands of the Body
Raymond You
Linear Algebra II
Ivan T. Ivanov
5/19/2013
1
What is Inertia?
Inertia is defined as an object’s resistance to changes in its state of motion. In this
analysis, the text will be mainly focused on the moment of inertia. Similarly to linear inertia,
moment of inertia is the measure of an object’s resistance to angular acceleration, specifically the
resistance to rotation.
Background
The Structure of the Hip
The hip usually refers to an anatomical region or a joint. It is situated lateral to the gluteal
region and inferior to the iliac crest. The innominate bone (or hip bone) is a large flat bone,
forming the lateral half of the pelvis. It is composed of three bones fused together, usually by
adulthood. It makes up part of the hip region. The two innominate bones form the bony pelvis
contributing to the proximal articular surface of the hip provided by the acetabulum. On the sides
of the innominate bone, the ilium, ischium and pubis forms a socket to hold the femoral head of
the femur. The femur, the largest bone in the body, is divided into the head, neck and shaft.
In order to understand a kick, only the proximal end of the femur will be studied. The
femoral head is about two-thirds a sphere and is mostly congruent to the socket lateral to the
innominate bone. The congruency is meant to reduce load strain and increase contact area
between the femoral head and the socket in order to reduce the force per area (stress). The hip
joint is also known as the acetabulofemoral joint which closely resembles a ball-and-socket
model. The primary function of the hip joint is to stabilize the body in static or dynamical
situations. The hip joints are supported by the synovial capsule, attached to the bony pelvis. The
bands of fiber run parallel to the femur. The hip joint capsule (the capsular ligament) is located
proximally on the acetabulum and intertrochanteric crest. To further reinforce the capsule, three
longitudinal bundles of fibers are wrapped around the capsule: the iliofermoral, ishiofemoral and
the pubofemoral ligaments (Oatis 648).
In a simplified case, the moment of inertia of the kick will mainly have an effect on the
capsule and the ligaments of the hip. A soccer kick will cause the femoral head to rotate in the
socket effectively putting strain on the surrounding fibrous bands.
A soccer kick is described as a proximal-to-distal motion of the kicking leg i.e., the leg
moving away from the body. The motion can be broken down by segmental and joint movements
existing on multiple planes. The backswing of the thigh will initiate the kick. The thigh will
experience a deceleration due to the shanks and the hip muscles. The main muscles involved in
the forward kick are the quadriceps and knee extenders, along with the longitudinal and
circumferential fibers that hold the femoral head into the socket. The forward acceleration of the
kicking leg is provided by the knee extender as well as a motion-dependent moment from the
quadriceps (Kellis and Katis 154-155).
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The instep form is by far the most effective method to produce a powerful kick. During
the kicking phase, the leg is drawn backwards and the hip extends by 29°. The hip rotates
externally in a forward motion following by the knee flexing. The forward motion is then led by
the rotation of the pelvis. The hip will also begin to flex as it rotates externally (Kellis and Katis
154). The analysis will however treat the kick as start and end points where the initial and final
moments of inertia are obtained through the study. Furthermore, the amount of work done will
be determined by the average force produced from the initial and final moments of inertia. The
contact between the foot and ball consumes about 15% of the energy produced from the motion
(Crosby). Nevertheless, to simplify the situation, the energy consumption will be neglected. To
perform the kick, a great amount of force is applied on the muscles responsible for the
movement. Therefore, a certain amount of energy must be supplied from the body in order to
establish this motion. This analysis will seek to determine how much energy the body needs to
provide per kick.
Fig.1.1. The femoral head in the socket is lateral to the innominate bone
supported by longitudinal bundles of fiber.
The soccer kick can be divided into 6 frames, where the body is posed in certain
positions. For this analysis, frames 3 and 6 are of interest. Position 3 is the position right before
the forward motion of the legs. Position 1 and 2 are the approach and wind up of the leg for the
full backswing. Frame 6 is the extension of the knee, approximately parallel to the horizontal
line. It is the follow-through of the kick.
3
Fig.1.2. The six positions of a soccer kick, where the main position of interest in
this analysis is frame 3 and 6. Frame 3 is the final position of the backwards wind up, and frame 6 is the final position of the
forward swing.
It should be noted that this is the coordinate system that will be followed.
Fig.1.3. The coordinate axis system throughout the investigation
Analysis of the Inertia of a Soccer Kick from Initial Position
To begin the analysis, the first frame of the motion will be as depicted a simplified image
demonstrating position 3 of Fig1.2.
Fig.1.4. The initial position according to the analysis. This is the transition
between the backswing of the leg to a forward swing; swing-limb loading.
The moment of inertia of a body is calculated by summing the product of mass and
distance squared for every particle in the body with respect to the axis of rotation.
4
The moment of inertia is given by
,
(1-1)
where m and r2 are the mass and distance squared with respect to the axis of rotation (the center
of mass of the body).
Moment of inertia is also dependent on the magnitude and distance of each point masses
relative to the axis of rotation. The further away particles are from the axis of rotation, the
greater contribution it’ll have to the overall moment of inertia. In the case of a rigid body, the
further away any body segment is from the center of rotation, the greater the segment will
contribute to the moment of inertia with respect to the center of mass.
In this analysis, inertia matrices will be used with the individual unit vectors to determine
the inertia dyadic. The different body segments are rotated with respect to the hip. The parallel
axis theorem is then used to determine the moment of inertia of the rigid body in reference to the
chosen axis of rotation.
The parallel axis theorem for moment of inertia about a new axis of rotation is given by
,
(1-2)
where Icm is the moment of inertia at the center of mass of the body segment in study, m is the
mass of segment and r is the perpendicular distance from the axis of rotation.
Information for Icm is obtained from “Principles of Biomechanics” where the moment of
inertia of the segment in study has already been determined. The study will focus on determining
the second moment of inertia; mr2.
The inertia dyadic is determined relative from point P (the segment of the body) relative
to the origin O (the hip) for the direction of the unit vectors, given by
(
),
(1-3)
where ux,y,z are the individual unit vectors parallel to the x, y and z axes, rPG is the position
vector relating the position of P to the origin O and M is the mass of the body segment. IP/O is the
second moment of inertia.
Notice that this is the equivalent to Eq.1-1 for moment of inertia, except now it is
expressed as a matrix.
5
PG is the position vector relative from the origin O (hip) to the center of mass of the
segment in study, following an XYZ coordinate system. The leg is assumed to rotate about the Zaxis; the effective rotation is on the X and Y plane, where the leg follows an angle of 29° (Kellis
and Katis 154). The PG vector presents the position in its X, Y, and Z components.
PG=
The parallel axis theorem states that the moment of inertia of a rigid body can be
determined in reference to the body’s center of mass. Following Eq.1-3, the perpendicular
distance r multiplied by the vector PG will result in the vector that can be used in order to
determine the position of the body segment in 3-dimensional space with the relative distance
from the origin O to the center of mass of the segment; rPG is the distance vector.
=
Following this, the distances in Ux, Uy and Uz can be determined by breaking down the
direction vectors and taking the cross products with the vector rPG.
ux=
uy=
uz=
6
(
)
The vector rPG appears twice which accounts for the r2 in Eq.1-1 and Eq. 1-2.
The parallel axis theorem will correct the moment of inertia that is distal from the center
of mass. The moment of inertia at the center of mass of the body must be added to the second
moment of inertia produced by the segment a distance away from the axis of rotation as indicated
by Eq.1-2.
ITotal= Icenter of mass+mr2
=
+
=
Notice that the resulting moment of inertia matrix is positive and symmetrical. Real and
positive eigenvalues are expected. Through an eigenvalue analysis, the values were
λ1= 0.0705999999993217
λ2=0.123391812833638
λ3=0.0707918128329600.
The positive and real eigenvalues verifies the process, and the moment of inertia can be
interpreted with real physical sense.
A similar procedure is done for the lower leg and the foot. For the lower leg, the distance
from the center of mass is the entirety of the upper leg and half the lower leg. The foot will
subsequently follow, where the distance is the entire upper and lower leg.
7
For the lower leg, the added distance of the upper leg will be incorporated in the distance
vector.
(
)
(
)
The unit vector in the y-direction implies that the rotation of the lower leg only occurs in
that specific axis.
The procedure to determine the second moment of inertia is the same for the lower leg as
for the upper leg, only differing in the new rPG vector.
The resultant inertia matrix is:
=
=
The moment of inertia matrix is positive and symmetrical and similarly, positive and real
eigenvalues are expected. Following an eigenvalue analysis, the following results were yielded.
λ1= 0.0502694938574550
λ2= 0.00589999999654500
λ3= 0.0451694938540000
The positive and real eigenvalues verifies the process.
To determine the next distance vector for the foot referring to the center of mass, both the
length of the upper and lower legs must be added together.
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=
The results for the foot are as follows:
=
=
The eigenvalue analysis resulted in positive and real values.
λ1= 0.0715613286919120
λ2= 0.00348931427908802
λ3= 0.0750506429710000
Again, the process to determine the second moment of inertia for the foot is exactly the
same for the upper and lower leg, only differing in the distance vector rPG.
The total moment of inertia produced from the initial position would be the sum for all
the moment of inertia produced by each body part distal from the center of rotation.
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=
Eigenvalues:
λ1= 0.236310447699954
λ2= 0.0867926163750063
λ3= 0.188903064074960
The eigenvalues were real and positive, verifying the procedure.
Next is to determine the rotation about the axis. The leg is assumed to mostly rotate in the
z-axis with a slight rotation in the y-axis as the motion of the kick is not completely rigid.
Assume that rotation in the y-axis is about 10°.
The vector defining this rotation of the leg will be given by,
V=
And therefore the rotation about the axis is determined by
10
Iv, v=
= 0.1612
.
Analysis of the Inertia of a Soccer Kick from Final Position
Fig.1.5. The final position according to this analysis; this is known as the followthrough phase of the kick.
Similar calculations are done with the final position of the kick. However, the PG position
vector changes. This is due to the kicking leg being parallel to the x-axis. The position vector
becomes:
PG=
By changing the position of the kicking leg, the moment of inertia produced by the upper
and lower leg and the foot will be equal to the sum of each individual inertial in relation to the
distance from the origin.
After similar calculations, the moment of inertia of the final position was determined.
ITotal for Final Position=
11
Positive and real eigenvalues verifies the procedure.
λ1= 0.0773000000000000
λ2= 0.282251050000000
λ3= 0.225351050000000
To Determine the Moment of Inertia for the Rotation about the Axis
The same procedure is used for the final moment of inertia as for the initial moment of inertia.
Iv,v=
= 0.2801
Following this, converting
to
to SI units will facilitate the conversion of moment
inertia into energy.
Since 1
is equivalent to 157.087464
then the initial position will have,
=
=25.33
And the final position will have,
=
=44.00
.
Once these results are obtained, conversion to torque will require the angular acceleration.
12
Angular acceleration
for angular acceleration is
is defined as the rate of change of angular velocity. The SI unit
.
To determine the angular acceleration, the rotation of the hips must be determined in
radians. To do that, the distance it travels in radians must be determined. Due to the motion of
the kick being a rotation, the toe-to -toe distance travelled from the initial to final position will
give an indication of the angular acceleration.
Fig.1.6. The black figure is the initial
position of the kick, whereas the red figure is the final position of the kicking leg. The rotation was nearly 180°. From empirical
observations, the angle from the toe in the initial position is approximately 20° below the horizontal axis. The toe-to-toe distance
travelled is determined to approximately be 3.17789 m.
The radius of the circle will be approximately the length of the leg; 1.138m. With the
known angle of the circular path, the arc length can be determined. This will be important in
finding the energy done by the motion.
This information will be valuable later.
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To establish the angular acceleration of this kick, the duration is also need. Assume a
kick on average lasts for 0.66 seconds.
Determining Angular Acceleration
= 6.41
Since torque is
,
then the torque produced from the initial frame is
) (6.41
)
) (6.41
)
=162.40
and that produced for the final position is
=282.08
.
It is assumed that the longitudinal fibers in the hips are about 15 cm long. The force can
be determined on the muscles responsible for the action.
Initial Force
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Final Force
To find the average force, half the sum of the initial and final forces will give the force
required to rotate the motion of the kick from the initial to final position.
To determine the energy required, work must be ascertain. Work is defined as the product
of force by distance.
Work Done by the Motion
Concluding Remarks
In order to determine how much energy the body requires to perform this motion, a
comparison is made with food calories to the energy in joules needed for the kick. On nutritional
information panels, the energy provided by the food is given in calories. This is misleading as the
units are actually kilocalories. The word ‘calorie’ is capitalized in some cases to denote 1 kcal
(Layton). For instance, a jellybean usually has 4.1 Calories (caloriecount.com), but in reality that
is 4.1 kcal or 4100 calories. In SI units, a calorie is equivalent to 4.148 Joules.
If the kick produced
of energy, the conversion to kilocalories would be
15
Then, 1.125 kcal in terms of the label on the food products would approximately be 1
calorie.
The body will convert food to metabolic fuel via various energy pathways. The main
source of energy used by the body is ATP (adenosine triphosphate). ATP is usually converted
from macromolecules such as carbohydrates, proteins and fat obtained through eating; however
carbohydrates will generally be the main source of fuel. In order to synthesize ATP, the
macromolecules will undergo an aerobic (using oxygen) or anaerobic (without oxygen)
respiration. There are two sub-categories of the anaerobic pathway known as the ATP-CP
anaerobic energy pathway and fermentation. The ATP-CP energy pathway supplies energy for
about 10 seconds by using the ATP stored in muscles (gives 2-3 seconds worth of energy) and
CP (creatine phosphate; gives 6- 8 seconds of energy) to make more ATP. It is mainly used for
explosive and quick motions. It does not require oxygen. Once ATP from the muscles and CP are
depleted, the body will either move to fermentation or aerobic respiration. Fermentation enables
the muscle cells to produce small amounts of ATP through glycolysis in the absence of oxygen
for short and high intensity burst of physical activity. In fermentation, pyruvate (a product of
glycolysis) will be reduced to lactic acid. The partial breakdown of glucose by fermentation will
only be sustainable until the lactic acid builds up significantly in the muscle, known as the lactate
threshold. Lactic acid accumulation causes muscle fatigue, pain and a burning sensation
noticeable within several minutes. Aerobic respiration will mainly fuel long lasting activities,
and use oxygen to make ATP as opposed to the other two forms of energy pathways (Quinn).
Kicking is mainly an anaerobic procedure, notably dependent on the ATP-CP energy
pathway (Crosby). The motion is very short lived and is usual a sudden movement requiring
short bouts of energy to follow through the motion. Considering that a soccer player may be
running and performing other physical activity for extensive period of time, the body will require
a steady source of energy. It will draw the energy required from the aerobic system as opposed to
anaerobic. If a jelly bean were to be consumed, the useful nutrient such as glucose could undergo
either aerobic or anaerobic respiration to produce a usable form of energy depending on the
situation. Glycolysis is the first step in extracting the energy from the jellybean. The glucose
undergoing glycolysis will produce pyruvate which can either lead to anaerobic respiration (in
the absence of energy) or will follow with the Citric Acid cycle (in the presence of oxygen).
Which pathway the pyruvate undergoes is largely dependent on the condition of the body.
In principle, one jellybean should provide enough energy to go through with at least 3
soccer kicks if only the energy provided is considered. In reality, the nutrients received from the
jellybean may be treated in several of ways and will ultimately be useful to the body one way or
another.
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Work Cited
Crosby, J.. N.p.. Web. 19 May 2013.
<http://www.sportsinjurybulletin.com/archive/biomechanics-soccer.htm>.
"Calories in Candies, Jellybeans." Calorie Count. N.p., n.d. Web. 19 May 2013.
Huston, Ronald L. Principles of Biomechanics. Boca Raton: CRC, 2009. Print.
Kellis, Eleftherios, and Katis, Athanasios. "Biomechanical Characteristics and
Determinants of Instep Soccer Kick." Journal of Sports Science and Medicine 6 (2007): 154-65.
Print.
Layton, Julia. N.p.. Web. 19 May 2013. <http://science.howstuffworks.com/life/humanbiology/calorie.htm>.
"Moment of Inertia." Moment of Inertia. N.p., n.d. Web. 11 May 2013.
Oatis, Carol A. Kinesiology: The Mechanics and Pathomechanics of Human Movement.
2nd ed. Philadelphia: Lippincott Williams & Wilkins, 2009. Print.
Quinn, E.. N.p.. Web. 1 Jun 2013.
<http://sportsmedicine.about.com/od/sportsnutrition/a/Energy_Pathways.htm>.