This senior project report models and compares the accelerations experienced by the skull within a traditional football helmet and the Vicis Zero1 helmet during impacts. The report:
1) Models simplified sections of each helmet as multiple two-mass spring systems, with the outer shell/padding represented as damped springs.
2) Derives the damping force equations for these systems, showing the force is proportional to the relative velocities of the two masses.
3) Attempts to numerically solve the models in Mathematica to compare skull accelerations, but encounters an error that prevents an accurate comparison.
1. Department of Physics and Astronomy
Ithaca College
Senior Project Report
Revolutionizing Helmet Designs
Reducing Risk of Concussions
Submitted by,
Schnayder Termidor
May 11, 2016
2. ITHACA COLLEGE DEPARTMENT APPROVAL
of a Senior Project submitted by
Schnayder Termidor
This senior project report has been reviewed by the senior project instructor and has been
found to be satisfactory.
Dr. Matthew C. Sullivan, Senior Projects Instructor Date
I understand that a digital copy of my senior project report will remain on file in the Depart-
ment, and may be distributed within the Department or College for educational purposes.
My signature below authorizes the addition of my report to this repository.
Schnayder Termidor Date
3. Abstract
In this project I attempted to model accelerations experienced by the skull
within sections of the traditional design for a football helmet and a new de-
sign, implemented by a company named Vicis, that involves the deformation
of the outershell of the helmet. This design, called the Zero1, was implemented
to address the issue of concussions, and can be simplified into multiple two
overdamped spring-coupled masses systems. These programs were designed
in Mathematica. Despite the programs passing preliminary tests, an error oc-
curs when attempting to implement the proportionality between the traditional
football helmet and the Zero1 helmet models. This error prevented an accu-
rate comparison of the accelerations experienced by the skull within these two
models.
i
5. 1. Introduction
Concussions have been the major topic of a widespread debate on the brutality of Amer-
ican football. Despite the intensive research on concussions, they are actually difficult to
define due to the lack of objective clinical and radio-graphic findings in routine imaging of
the brain. [5] However, what is known is that concussions are induced by traumatic bio-
mechanical forces such as a head-to-head hit in American football. [5] Current research has
drawn connections between concussions and the debilitating disease chronic traumatic en-
cephalopthy.
Chronic traumatic encephalopathy is a form of neurodegeneration that results from repet-
itive brain trauma. Symptoms of CTE may begin years or decades later and include a pro-
gressive decline of memory, as well as depression, poor impulse control, suicidal behavior,
and eventually, dementia similar to Alzheimer’s disease. [4]
Recent headlines have consisted of ex-National Football League players suffering from
these symptoms and taking their own lives as a result. One would assume that the NFL
would attempt to reduce these incidents from occurring. Yet, the NFL was recently accused
of attempting to hide information that linked football-related head injuries, such as concus-
sions, to permanent brain injuries like CTE. Despite this, American Football and the NFL
are still very much a part of American culture, with no noticeable reduction in the number
of athletes that aspire to play at the professional level. Many of the current athletes in the
NFL used football to escape environments with no promising future. Playing professional
football may have been the only way for these players to support themselves and their fam-
ilies. These athletes are also seen as heroes, and many children start playing football to
emulate them. Combine these facts with the amount of revenue generated by the NFL, ($8.9
billion in 2013), and the immense love for the game by most of the general public, and the
removal or cancellation of American Football seems improbable. [3]
Football helmets were originally introduced to reduce the risk of skull fractures in players.
1
6. Introduction
Concussions were not as prevalent when the original hard shell helmet was designed. Yet,
the hard shelled helmet is the base design for all helmets that are currently in the market;
see Fig.1.
Hard Outer Shell Foam Padding
Figure 1: Cross section of traditional helmet that was designed to reduce skull fractures.
The hard outershell works to disperse forces during an impact. The primary function of the
foam padding is comfort, but it also absorbs some forces during an impact. [1]
Once concussions began to become more prevalent, helmet manufacturers began to mod-
ify the original helmet design to address the issue of concussions.For example, the Xenith
helmet design implements shock absorbing air-filled discs that compress and release air, and
then quickly re-inflates in response to an impact. [7] In order to determine the effectiveness
of helmets, tests such as the Head Impact Telemetry System and the Virginia Tech Ratings
were implemented. These tests measure a helmet’s ability to modulate the energy transfer
from an impact to the head, resulting in lower head accelerations. Lower head accelerations,
2
7. Introduction
in turn, reduce the risk of concussions. [9] Although the Xenith helmet implemented this new
technology, it still received lower ratings than helmets that use the traditional foam or air
padding on the Virginia Tech rating test. [7] Although a wide variety of new helmet designs
have been introduced in an attempt to reduce concussions, the amount of concussions diag-
nosed in the NFL in 2015 was 271, which is nearly a 32 percent increase from the previous
year. [8]
Vicis is a company that is currently attempting to revolutionize the design of the foot-
ball helmet by taking the design away from traditional methods. Chief medical officer of
Vicis claims that this occured because “current helmets were never intended to deal with
concussions. To take a product that is built for one purpose and try to retrofit it to address
concussions is a very challenging task." [11] This company has engineered a helmet called the
Zero1 that implements these non-traditional designs. “Unlike traditional helmets, the Zero1
has a soft outer shell that deforms on impact and a column-like inner structure intended to
absorb impact and disperse its force omnidirectionally." [11] This design is shown in Fig. 2.
The concept of the helmet is based on what the Virginia Tech Ratings measure. “If force
is the product of mass and acceleration and you can’t change the mass of a football player,
a helmet must address acceleration." [2] By implementing designs that address Newton’s
second law of motion, Vicis claims that it can reduce the amount of force a players head will
experience during an impact; which may aid in reducing the amount of concussions in all
levels of football.
Vicis is revolutionizing the design of the football helmet in order to combat the arising
trend in ex-NFL players suffering from CTE and prevent the damages that this disease
inflicts on players and their families. The deformable outer shell will reduce the acceleration
experienced by the skull in ways that the traditional helmet cannot, thus reducing the amount
of concussions sustained by players. Reducing the risk of sustaining concussions during a
game of football in return will make the sport a lot safer than it currently is.
Although this new design seems to be fantastic at tackling the issue of concussions, it
3
8. Introduction
Form Liner
Waterproof textiles and
foams create a form liner
that mimics mattress-like
memory foam for fit and
comfort.
Arch Shell
Hard plastic layer
protects against skull
Chin Strap
Two of the four snaps
fasten to the inner shell to
curb energy flowing
through the jaw.
Lode Shell
Softer shell absorbs
impact load by deforming
like a car bumper, then
bouncing back.
Core Layer
Inch-and-a-half-thick
layer of vertical struts that
bend and buckle to slow
down impact forces.
The Helmet Reimagined
Source: Vicis
Figure 2: Cross section of Zero1 helmet that was designed by Vicis.The lode shell and core
layer were innovated in order to address the issue of concussions. The core layer works
to address rotational as well as linear forces, while the lode shell mainly addresses linear
forces. [2]
has yet to be subject to the Virginia Tech Ratings and Head Impact Telemetry System
tests. There is a possibility that the Zero1 helmet receives a lower grading than traditional
helmets just like the Xenith helmet did. Vicis’ claims can not be proven true until the Zero1
undergoes the standard testing for helmets.
Despite the Virginia Tech Ratings and Head Impact Telemetry tests being the ultimate
factor in deciding whether the Zero1 helmet will revolutionize the designs of football helmets,
there are other ways to determine the Zero1’s ability to modulate forces. With the use of
my knowledge of the bio-mechanics of a concussion from studying exercise science, and my
knowledge of analytical mechanics, I numerically modeled simplistic versions of the Zero1
helmet and the traditional helmet receiving an impulse. These models produced graphs
that allowed me to compare the linear accelerations experienced by the skull within these
helmets during impacts of varying magnitudes. Comparing the acceleration experienced by
4
9. Introduction
the skull within the Zero1 and the traditional helmet these allowed me to determine if Vicis’
new design of the football helmet can actually reduce the amount of force a player’s skull
will experience during an impact. If the acceleration experienced by the skull within the
Zero1 is of a lesser magnitude than the acceleration experienced in the traditional helmet,
players wearing the Zero1 helmet during practice and competition will have a reduced risk
in sustaining concussions. A reduced risk in sustaining concussions thus causes a reduced
risk is suffering from CTE.
5
10. 2. Theory
As shown in Fig. 1, the traditional helmet design consists of some type of extremely stiff
foam padding surrounded by a hard spherical like outer shell. During an impact, the outer
shell works to disperse forces while the foam padding absorbs some of the force. The Zero1
design consists of the same concept, (Fig. 2) , however the outer shell, also known as the
lode shell, is deformable. During an impact this allows the outer shell to “absorb the impact
load by deforming like a car bumper and then bounce back." [2] Given these qualities of the
padding of the traditional helmet, and the lode shell of the Zero1 it can be assumed that they
both behave like the system of masses connected via springs. When a spring is compressed
or extended, the mass experiences a restoring force that is proportional to the distance that
the spring is compressed or extended. [6] This force is shown in Eq. 1, and is also known as
Hooke’s Law.
m¨x = −kx (1)
where x is the distance the spring is compressed or extended, and k is the spring constant.
In a perfect spring system the mass would continue to oscillate at the same amplitude
until an external force acts on the mass to end the oscillations; this motion is known as
simple harmonic motion. However during an impact the outer shell of the Zero1 helmet
and the padding of the traditional helmet are compressed and then slowly return to their
equilibrium position without any noticeable oscillations. This means that rather than acting
as perfect springs, the outer shell of the Zero1 and the padding of the traditional helmet act
as damped springs and in fact, overdamped springs. Damped oscillators can be thought as
simple harmonic oscillators but with an exponentially decreasing amplitude. [10] The expo-
nential decrease in amplitude occurs due to the damping force. The equation of motion of a
damped spring is shown below in Eq. 2
6
11. Theory
m¨x = −kx − b ˙x (2)
where b is the damping constant of the damping force.
In order to simplify the code required to model the Zero1 and the traditional helmet,
sections of these helmets were modeled rather than the full helmets. The sections of these
helmets were modeled as springs coupled by masses, and is shown in Fig. 3.
Figure 3: Simplified versions of sections of the traditional football helmet and the Zero1.
These sections were simplified in order to observe the accelerations experienced by the skull
during an impact in Mathematica.
7
12. 2.1 Damping Force of Football Helmet Theory
2.1 Damping Force of Football Helmet
The damping force (−b ˙x) shown in Eq. 2 is proportional to the velocity of the mass on the
spring. This means that the damping force is at its greatest when the mass is traveling at
its max velocity. In the mass spring system, when the spring is compressed or stretched, the
damping force is strongest when the point mass is traveling at its maximum velocity, this
point is when the spring reaches its equilibrium length.
The models of the sections of the Zero1 and traditional helmet (Fig. 3) can be broken
down into multiple two spring-coupled mass systems given. An example of this system is
shown in Fig. 4.
X
x1
x2
x
Figure 4: Two spring-coupled mass system. m1 and m2 are the masses coupled by a spring.
Solid vectors denote the position of the masses in terms of the center of mass of the system.
The large X represents the the center of mass of the two masses while the smaller x represents
the length of the spring. The dotted vectors represents the positions of the masses relative
to each other. x1 is the position of m1 and x2 is the position of m2.
The motion occurring in this system can be defined in terms of the center of mass of the
coupled masses or in terms of the position of the masses relative to each other. The center
of mass for this system is defined as Eq. 3.
8
13. 2.1 Damping Force of Football Helmet Theory
X =
m1x1 + m2x2
m1 + m2
(3)
where X is the center of mass.
In terms of the center of mass motion, since there is no external force acting upon the
center of mass the equation of motion for the skull is:
¨X = 0 (4)
The motion of the coupled masses is dictated by the length of the spring x = x2 − x1.
When the spring is compressed or extended the masses experience a restoring force that will
cause the masses to oscillate. Since the spring is damped these masses will oscillate with
a decreasing amplitude. In terms of the relative positions the equation of motion for the
spring-coupled masses is shown in Eq. 5.
¨x = −ω2
relx − b ˙x (5)
where ωrel is the frequency of the oscillations of the two masses relative to each other
and −b ˙x is the damping force of the spring.
In terms of the relative position, −b ˙x is the correct form of the damping force. However, the
model that I am coding in Mathematica calls for the damping force in terms of the motion
of the masses m1 and m2. This means that the damping force should be dependent on the
relative velocities of the masses. Since x is the length of the spring in the equations of motion
in terms of the relative motion, and is defined as x2 − x1, the damping force can also be
defined as Eqs. 6 and 7.
γ( ˙x1 − ˙x2) (6)
9
14. 2.2 Traditional Helmet Model Theory
γ( ˙x2 − ˙x1) (7)
This means that the damping force in my model will be dependent on the relative veloc-
ities of the outer shell and skull ˙x1 , ˙x2. Although both damping forces will be dependent on
the relative velocities, the skull and outer shell will experience the same magnitude of force
but in opposite directions.
2.2 Traditional Helmet Model
The model of the skull within the traditional helmet was further simplified down to two
point masses connected by a spring as shown in Fig.5.
m1 m2
k
Figure 5: Additionally simplified model of the traditional football helmet design. m1 repre-
sents the skull while m2 represents the outershell. Simplifying the traditional football helmet
model to this extent allowed for a simpler program to be written without altering expected
results.
As long as the average force acting upon outer shell and skull is constant, the acceleration
experienced by the skull for varying magnitudes of force will be the same for both systems.
The equation of motion in terms of x and y position for the outer shell is shown in Eq. 8.
The equation of motion for the skull is shown in Eq. 9.
m1 ¨x1 = k( (x2 − x1)2 + (y2 − y1)2 − l) − γ( ˙x1 − ˙x2) (8)
m2 ¨x2 = −k( (x2 − x1)2 + (y2 − y1)2 − l) − γ( ˙x2 − ˙x1) (9)
Where x1 and x2 are the positions of the skull and outer shell respectively; k is the spring
10
15. 2.3 Zero1 Model Theory
constant of the springs; l is the equilibrium length of the springs; γ is the damping constant;
and ˙x1 and ˙x2 are the velocities of the skull and outer shell relative to each other.
Eqs. 8, and 9 cause the shell and skull to experience 3rd law pairs of forces in the x
direction when the point masses are moved away from their equilibrium positions, x1 and
x2, evidently causing the spring to stretch or compress. The damping force works to decrease
the amplitude and the amount of oscillations experienced by the point masses, recreating
the overdamped characteristics of the football padding.
Prior to an impact the outershell of the traditional helmet and the skull will be traveling
at a velocity v. The impact will stop the forward motion of the outershell while the skull
continues to travel at a velocity v until its motion is affected by the force from the compression
of the padding of the helmet. An impact can be modeled in my program by having mass
1 traveling at a velocity v and making mass 2 stationary by setting the equation of motion
(Eq. 8 )and initial velocity of point mass 2 equal to zero (Eq.10).
m2 ¨x2 = 0 (10)
The acceleration experienced by mass 1 will be the linear acceleration experienced by the
skull during an impact in the traditional helmet.
2.3 Zero1 Model
Modeling a section of the Zero1 involved coupling 9 springs in order to recreate the defor-
mation of the Lode shell during an impact. 5 springs represented the Zero1 Vertical Struts
while the lode shell was composed of 4 springs. As shown in Fig. 3, each spring in the lode
shell is connected to an adjacent one by a point mass. These point masses are attached to
the ends of the springs that compose the Vertical Struts. The other ends of the vertical
struts are attached to a rod that represents the skull.
Impacts that stop the forward motion of the point masses in the outershell can be mod-
11
16. 2.3 Zero1 Model Theory
eled by making the center of mass of the rod travel at some velocity v in the x direction
and having some combination of point masses of the lode shell remain stationary. In Fig.1
6 point mass 3 was designated to be stationary. This will cause all of the springs in the
model to move away from their equilibrium length and cause the point masses to experience
restoring forces in the x and y directions. This event recreates the deformation of the lode
shell during an impact in the Zero1 model.
m3
Figure 6: Model of a section of the Zero1 during an impact. Point masses 1, 2, 4 and 5 of
the Lode Shell as well as the center of mass of the skull are moving at a certain velocity v.
Mass 3 is stationary, modeling an impact that stops the forward motion of m3. This causes
the springs to move away from their equilibrium length and the point masses to experience
a restoring force. This event recreates the deformation of the lode shell during an impact.
The equation of motion of the skull in the Zero1 is shown in Eq. 11. The equation of
motion of the point masses in the lode shell is shown in Eqs. 12, 13, 14, 15.
12
18. 2.3 Zero1 Model Theory
mn ¨yn =
4
n=2
kI( (xn − X)2 + (yn − Yno)2 − lnI)
(yn − YnO)
(xn − X)2 + (yn − YnO)
− γI( ˙yn − ˙Y nO)
+
4
n=2
kO( (xn+1 − xn)2 + (yn+1 − yn)2 − lnO)
(yn+1 − yn)
(xn+1 − xn)2 + (yn+1 − yn)
− γO( ˙yn − ˙yn+1)
−
4
n=2
kO( (xn − xn−1)2 + (yn − yn−1)2 − ln−1O)
(yn − yn−1)
(xn − xn−1)2 + (yn − yn−1)
− γO( ˙yn − ˙yn−1)
(15)
where X is the center of mass of the rod; ˙X is the velocity of the center of mass in the x
direction; Y nO are the positions where the springs are attached to the rod representing the
skull; mn is the mass of a point mass in the outer shell; xn and yn are their positions in the
x and y direction; kO and kI are the spring constants of the springs in the lode shell (outer
shell)l and vertical struts (padding) of the Zero1; lnO and lnI are the equilibrium lengths of
the springs in the lode shell and vertical struts of the Zero1; γI , and γO, are the damping
constant of the springs that represent the lode shell and vertical struts.
Analogous to the traditional helmet model, Eqs. 12, 13, 14,11 and 15 cause the point
masses and the center of mass of the skull to experience 3rd law pair of restoring forces
in the x and y directions when the point masses or the center of mass of the skull are
moved away from their equilibrium positions, X, x11, x2, x3, x4, and x5. The only exception
is the center of mass of the skull since it was only allowed to travel in the x direction
in my program. The damping force works to decrease the amplitude and the amount of
oscillations experienced by the point masses and the center of mass of the skull, recreating
the overdamped characteristics of the football padding.
Observing the accelerations experienced by the center of mass of the rod during impacts
of varying magnitudes in the x-direction aided in determining if the Zero1 can actually
modulate the forces of a linear impact better than the traditional design of the football
helmet.
14
19. 3. Experiment
The purpose of this project was to create programs in Mathematica that modeled simplified
versions of a section of the traditional football helmet, as well as a simplified version of a
section of the Zero1 helmet (Fig. 3). These programs produced graphs that allowed me
to compare the linear accelerations experienced by the skull within these helmets during
impacts of varying magnitudes.
Rather than modeling the traditional football helmet and the skull as two linear masses
connected by multiple damped springs, I modeled them as two point masses coupled by a
single overdamped spring. (Fig. 5). This allowed me to simplify the code for the traditional
helmet program since both systems are expected to behave the same as long as the same
force per mass is applied.
3.1 Point Masses
Each point mass that was used to represent the outer shell and the skull corresponded with
actual values of the mass of an average adult male skull, and the mass of a football helmet.
The most common type of helmet used by players in 2015 was the Riddell Revolution Speed.
This helmet weighs about 4lbs. [12] Since 1 lb = 4.448 22 N, the weight of the Revolution
Speed helmet converted to Newtons was calculated to be 17.79 N. Using the equation for the
force due to gravity on Earth’s surface (Fg = mg) the mass of the helmet was calculated to
be 1.81 kg. Since I am only modeling a section of the skull and the helmet, in the traditional
helmet model the point mass that represents the outer shell was designated to have a mass
of 0.241 032 kg or 22% of 1.81 kg . In the Zero1 model, the point masses of the outer shell
15
20. 3.2 Spring and Damping Constants Experiment
had masses that were proportional to the number of point masses in the outer shell 16
m =
1
n
0.241 kg (16)
Where m is the mass of the point masses in the outer shell and n is the number of point
masses in the outer shell.
"An adult human head cut off around vertebra C3, with no hair, weighs somewhere
between 4.5kg and 5kg." [13] This means that any mass between 4.5 kg and 5 kg can be used
to represent the skull in both models. For both programs a mass of 1.0956 kg or 22% of
4.98 kg was used.
The initial positions of the point masses were dictated by the equilibrium lengths of the
springs. The length of the padding of the football helmet will determine the x-position of
the point masses, while the length of the springs in the Lode Shell will determine their y-
position. Since the length of the padding varies from helmet to helmet within a range of
approximately 4 cm to 8 cm, the equilibrium length chosen for the inner springs lnI is 5 cm
or 0.05 m. The equilibrium length chosen for the springs of the outer shell (lnO) is 10 cm or
0.2 m.
3.2 Spring and Damping Constants
Since I was unable to find the exact spring and damping constants of the padding of the
traditional helmet and the outer shell of the Zero1, I had to approximate these values based
on the physical characteristics of football helmets. The padding of a football helmet consists
of extremely stiff vinyl foam, or other extremely stiff substances. In other words, the spring
constant of the padding of the traditional helmet is expected to be large. During an impact
the padding of a football helmet compresses and then slowly returns to its original position
with no noticeable oscillations to the naked eye. This means that the padding must be over-
damped or have a large damping constant. The lode shell of the Zero1 behaves similarly to
16
21. 3.3 Tests to Determine Program Functionality Experiment
the padding of the traditional football helmet, thus the damping and spring constants are
expected to be large as well. [10]
3.3 Tests to Determine Program Functionality
The programs that model the traditional helmet and the Zero1 allowed me to apply an
impulse to the outer shell of the helmets. By making the outer shell stationary and varying
the velocity that the skull is traveling, impulses of varying magnitudes to the outer shell
were modeled.
Both programs need to demonstrate a loss of energy in the oscillations of the outer shell
and skull that is based on their relative speeds to one another. As a result of this both
programs should be able to model the skull and outer shell of the helmet traveling at the
same velocity without any oscillations occurring. Correspondingly, if the outer-shell and
skull are given an initial velocity and the outer shell is given an initial position away from
its equilibrium position, both should oscillate and come to a rest without altering the initial
velocity of the whole system.
In order to determine if the Zero1 program is functioning properly, the program should
allow me to alter the spring and damping constants of the springs in the lode shell as well as
the core layer of the Zero1 model. If the spring and damping constants in the lode shell are
large the acceleration experienced by the skull should match the acceleration of the point
mass that represents the skull in the two point mass spring system. Raising the spring and
damping constants of the springs in the lode shell to a large number makes the outer shell
of the Zero1 stiff, just like in the design of the traditional helmet.
Rather than immediately creating the Zero1 model with 5 point masses in the outer shell,
a 3 point mass model was used in order to simplify the model for tests. Once the 3 point
mass Zero1 model met the criteria more point masses would be included in the model.
Once the programs met the criteria, the damping constants, spring constants, and initial
velocities were varied in order to compare the accelerations experienced by the skull within
17
22. 3.3 Tests to Determine Program Functionality Experiment
the Zero1 and the traditional football helmet.
18
23. 4. Results
In order to successfully compare both the Zero1 and traditional helmet’s ability to decrease
the linear accelerations experienced by the skull, the programs must first pass the tests de-
scribed in Section 3.3.
4.1 Test Results
When both the outer shell and skull are given an initial velocity of 5 m
s
the point masses
should continue to travel at this velocity without any oscillations. The only time there should
be oscillations is when the point masses experience the forces of the spring when they are
moved away from their equilibrium positions. The forces experienced by the point masses
are dependent on the positions and velocities of the point masses relative to each other. The
testing of both programs is shown in Figs. 7 (Traditional) and 8 (Zero1).
Correspondingly, if the center of mass of the skull and the point masses are given the same
initial velocity but are moved away from their equilibrium positions, they should oscillate
and then return to traveling at their initial velocity. The graphs of the programs undergoing
the tests are shown in Figs. 10(traditional), 12, and 11 (Zero1). Each point mass was moved
0.05 m.
19
24. 4.1 Test Results Results
Figure 7: Graph of Position vs. Time for point mass 1 in the traditional helmet model. This
graph is a result of the test to determine if the point masses will remain traveling at the
same velocity with no oscillations when given the same initial velocity.
Figure 8: Graph of Position vs. Time for the center of mass of the skull in the x direction,
point masses 1, 2, and 3 in the Zero 1 model. This graph is a result of the test to determine
if the point masses and center of mass of the skull will remain traveling at the same velocity
with no oscillations when given the same initial velocity.
20
25. 4.1 Test Results Results
Figure 9: Graph of Position vs. Time for the center for the center of mass of the skull, point
masses 1, 2, and 3 in the y direction of the Zero 1 model. This graph is a result of the test
to determine if the point masses and center of mass of the skull will remain traveling at the
same velocity with no oscillations when given the same initial velocity.
Figure 10: Graph of Position vs. Time for point mass 1 in the traditional helmet model. Point
mass 1 and point mass 2 were given an initial velocity of 5 m
s
and point mass 2 (outershell)
was displaced away from its equilibrium positions by 10 cm in the positive x direction. This
graph is a result of the test to determine if the point mass will return to traveling at its
initial velocity after being displaced away from its equilibrium position.
21
26. 4.1 Test Results Results
Figure 11: Graph of Position vs. Time for the center of mass of the skull, point masses
1, 2, and 3 of the Zero 1 model in the x direction. The center of mass of the skull, point
masses 1, 2, and 3 were given an initial velocity of 5 m
s
. The point masses in the lode shell
were displaced away from their equilibrium positions by 10 cm in the positive x direction.
This graph is a result of the test to determine if the point masses will return to traveling
at their initial velocities in the x direction after being displaced away from their equilibrium
position.
22
27. 4.2 Raising Spring and Damping Constants of Lode Shell Results
Figure 12: Graph of Position vs. Time for point masses 1, 2, and 3 of the Zero 1 model
in the y direction. The center of mass of the skull, point masses 1, 2, and 3 were given an
initial velocity of 5 m
s
. The point masses in the lode shell were displaced away from their
equilibrium positions by 10 cm in the positive x direction. This graph is a result of the test
to determine if the point masses will return to thier original y-position after being displaced
away from their equilibrium position in the x-direction.
4.2 Raising Spring and Damping Constants of Lode Shell
When raising the spring and damping constants of the outer shell of the Zero1 to a large
number, the outer shell of the Zero1 is no longer allowed to deform. This means that if the
average force acting upon the outer shell of the Zero1 and the traditional helmet are equal,
the skull should experience the same magnitude of acceleration within each helmet.
The spring and damping constant of the padding of the traditional helmet was chosen to
be 850 N
M
and 90 Ns
m
respectively. The point mass representing the skull was given an initial
speed of 5 m
s
and the second point mass was made to be stationary. The skull’s (point mass
1) position over time is shown in Fig. 13. The skull’s velocity is shown in Fig. 14.
In order to recreate the acceleration experienced by the skull within the traditional helmet
model in the Zero1 program the spring and damping constants of the vertical strut section
were designated to be 283 N
M
and 30 Ns
m
due to the proportionality between the traditional
23
28. 4.3 Analysis: Large Spring and Damping Constants of the Lode Shell Results
Figure 13: Graph of x Position vs. Time for point mass 1 in the traditional helmet model.
Spring constant and damping constant of padding are 850 N
M
and 90 Ns
m
respectively. Point
mass 1 was given an initial velocity of 5 m
s
while the second point mass was made to be sta-
tionary to model an impact that stops the forward motion of the outershell of the traditional
helmet.
helmet model and the Zero1 model. The spring constants and damping constants of the
Lode Shell were designated to be 1 000 000 N
M
and 10 000 Ns
m
respectively. Point mass 2 was
stationary, while point mass 1, 3, and the center of mass of the skull were given an initial
velocity of 5 m
s
. The acceleration experienced by the center of mass of the skull is shown in
Fig. 15.
4.3 Analysis: Large Spring and Damping Constants of
the Lode Shell
Figs. 16 and 15, reveal that there is some sort of error in the Zero1 program. The spring
and damping constants of the outer shell were raised to a large number, while the spring and
damping constants of the padding were designated based on the proportionality of the Zero1
model and the traditional helmet model as shown in Eq. 16. The point masses are expected
to remain stationary while the skull’s position, velocity and acceleration resembles that of
24
29. 4.3 Analysis: Large Spring and Damping Constants of the Lode Shell Results
Figure 14: Graph of Velocity in the x direction vs. Time for point mass 1 in the traditional
helmet model. Spring constant and damping constant of padding are 850 N
M
and 90 Ns
m
re-
spectively. Point mass 1 was given an initial speed of 5 m
s
while the second point mass was
made to be stationary to model an impact that stops the forward motion of the outershell
of the traditional helmet.
Figure 15: Graph of Velocity in x vs. Time for the center of mass of the skull in the
Zero1 model. Spring constant and damping constant of vertical strut section are 283 N
M
and
30 Ns
m
respectively. The spring and damping constants of the lode shell are 1 000 000 N
M
and
10 000 Ns
m
respectively to make the lode shell stiff. Point mass 1 and 3 as well as the center of
mass of the skull were given an initial velocity of 5 m
s
while the second point mass remained
stationary.
25
30. 4.3 Analysis: Large Spring and Damping Constants of the Lode Shell Results
Figure 16: Graph of Position vs. Time for point masses 1 and 3 as well as the center of
mass of the skull in the Zero1 model. Spring constant and damping constant of vertical strut
section are 283 N
M
and 30 Ns
m
respectively. The spring and damping constants of the lode shell
are 1 000 000 N
M
and 10 000 Ns
m
respectively to make the lode shell stiff. Point mass 1 and 3 as
well as the center of mass of the skull were given an initial velocity of 5 m
s
while the second
point mass is stationary.
the skull within the traditional helmet because of the high spring and damping constants
designated to the lode shell of the Zero1.
In order to further investigate the error in the Zero1 model, spring and damping constants
of the vertical strut section were set equal to their tradtional helmet counterparts. Spring
and damping constants of 850 N
M
and 90 Ns
m
respectively were used. The graph of the skull’s
velocity with the same parameters as the traditional helmet model is shown in Fig. 17. The
skull’s position as well as the positions of the point masses are shown in 18
Figs. 17, and 18 reveal that when the spring and damping constants of the vertical strut
section is set equal to the spring and damping constants of the traditional helmet padding,
the average acceleration experienced by the skull within the Zero1 is greater than the average
acceleration experienced by the skull within the traditional helmet model. The position of
point masses 1, and 3 in the lode shell remain stationary with point mass 2 despite giving
them both an initial velocity of 5 m
s
. This behavior is expected, and is what I sought to
26
31. 4.3 Analysis: Large Spring and Damping Constants of the Lode Shell Results
Figure 17: Graph of Position vs. Time for the center of mass of the skull in the Zero
1 model. Spring constant and damping constant of vertical strut section are 850 N
M
and
90 Ns
m
respectively. The spring and damping constants of the lode shell are 1 000 000 N
M
and
10 000 Ns
m
. Point mass 1 and 3 as well as the center of mass of the skull were given an initial
velocity of 5 m
s
while the second point mass remained stationary.
Figure 18: Graph of Position vs. Time for the center of mass of the skull in the Zero
1 model. Spring constant and damping constant of vertical strut section are 850 N
M
and
90 Ns
m
respectively. The spring and damping constants of the lode shell are 1 000 000 N
M
and
10 000 Ns
m
. Point mass 1 and 3 as well as the center of mass of the skull were given an initial
velocity of 5 m
s
while the second point mass remained stationary.
27
32. 4.3 Analysis: Large Spring and Damping Constants of the Lode Shell Results
include in the Zero1 model. In the case of having 3 point masses in the lode shell, when the
proportionality between the Zero1 and traditional helmet models are not taken into account,
setting the spring and damping constants of the two models equal will cause the skull to
experience 3 times the amount of force in the Zero1 model when compared to the traditional
helmet model.
Despite the Zero1 model behaving as expected, for some reason when the proportionality
of the spring and damping constants of the models are taken into account errors occur. These
errors are shown in Figs. 15, and 16.
28
33. 5. Conclusions
Concussions have been the topic of a major widespread debate on the brutality of American
football. Many companies have been working towards reducing the risk of players sustaining
concussions while playing American football. Vicis is a company that is attempting to rev-
olutionize helmet designs by creating a helmet with a deformable outer shell. This helmet
is called the Zero1 and it has yet to undergo the standard grading of a helmet’s ability to
reduce the magnitude of force experienced by the skull during an impact.
I attempted to model sections of both the traditional helmet design and the new de-
formable outer shell design in Mathematica in order to observe the accelerations experienced
by the skull during an impact; thus proving that Vicis’ new helmet design should become the
new standard for football helmets. Although the the Zero1 program and traditional helmet
program passed the preliminary tests. The program that models an impact to the Zero1
helmet did not behave as expected. Errors occurred when altering the spring and damping
constants to values based on the proportionality of the Zero1 and traditional helmet model.
This prevented me from comparing the accelerations experienced by the skull within these
helmets.
5.1 Future Investigation
Due to time constraints, I was unable to complete my senior project to the extent that I
would have liked. In the best case scenario, the Zero1 model would have behaved as expected
and would have allowed me to observe the acceleration experienced by the skull and then
compare it to the acceleration experienced by the skull within the traditional helmet.
In future investigations I would like to be able to create a program that aids in finding the
average acceleration experienced by the skull for a variety of different spring and damping
constants.
29
34. 5.1 Future Investigation Conclusions
The constants used in this project were based on the assumptions that a football helmet’s
padding is extremely stiff and overdamped. I would have liked to be able to investigate and
use real values for the damping and spring constants of football helmet padding.
The only type forces that were addressed in this project were linear forces. In future
experiments I would like to be able to fully model the helmets in order to aid in addressing
the rotational and linear accelerations experienced by the skull during an impact.
30
35. References Cited
[1] V. Bologna, N. Kraemer, R. Infusino, and T.M. Ide. Protective sports helmet, March 14
2013. US Patent App. 13/229,165.
[2] Peter Robinson Bryan Gruley. This football helmet crumples - and that’s good. http:
//www.bloomberg.com/features/2016-vicis-football-helmet/. Accesed: 2015-02-
04.
[3] E. Hunt. Ethics of a billion-dollar sports league. Baylor Business Review, 2015.
[4] Shaheen E. Lakhan and Annette Kirchgessner. Chronic traumatic encephalopathy: the
dangers of getting “dinged”. SpringerPlus, 1(1):1–14, 2012.
[5] H. J. McCrea. Concussion in sports. Sports Health, 2013.
[6] Randall. Physics for Scientist and Engineers: A Strategic Approach, chapter •, pages
488 – 489. Pearson, three edition, 2013.
[7] C. Smith. Hard knocks: Xenith’s helmet technology stands tall amidst football’s con-
cussion crisis. Forbes, 2014.
[8] Michael David Smith. Nfl says 271 concussions were diagnosed
in 2015. http://profootballtalk.nbcsports.com/2016/01/29/
nfl-says-271-concussions-were-diagnosed-in-2015/. Accesed: 2015-02-04.
[9] Med Staff. Football helmet design can reduce concussion risk. Medical Design News,
2014.
[10] John R. Taylor. Classical Mechanics, chapter 5. University Science Books, 1939.
[11] Abigail Tracy. Could this helmet save football from the sport’s concus-
sion problem? http://www.forbes.com/sites/abigailtracy/2016/02/04/
nfl-cte-football-concussions-injuries-helmet-vicis-zero1-super-bowl/
#30fb352559c5. Accesed: 2015-02-04.
[12] Sports Unlimited. How much does a football helmet weigh? Accesed: 2016-03-04.
[13] Danny Yee. Average human head weight. Accesed: 2016-03-04.
31
