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Roller Coaster Design
Connor Hain
April 25, 2016
Contents
1 Introduction 2
2 Circular Loop 6
2.1 Motion in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Energy Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Air Resistance/Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 High Air Resistance . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 Low Air Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 G-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Clothoid Loop 16
3.1 Defining the Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Motion Along a General Curve . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Motion Around the Clothoid Loop . . . . . . . . . . . . . . . . . . . . . 22
3.4 General Energy Argument . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.5 G-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.6 Air Resistance/Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Circular Helix Loop 29
4.1 Defining the Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Frenet-Serret Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Motion Around the Circular Helix Loop . . . . . . . . . . . . . . . . . . 32
4.4 G-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Clothoid Helix Loop 36
5.1 Motion Around the Clothoid Helix Loop . . . . . . . . . . . . . . . . . . 37
5.2 G-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Multi-Mass Problem 40
6.1 Two-Mass Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2 Three-Mass Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.3 N-Mass Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
7 Conclusion 47
1
1 Introduction
Roller coasters have been the headline attractions at theme parks for many years.
Whether they are dominating the skyline or boasting a large number of inversions, they
have progressed from being relatively simplistic machines, to becoming highly complex
structures, with incredibly complicated control systems.
The term ’roller coaster’ is thought to have originated from American thrill rides,
from the late nineteenth century. These rides used hundreds of rollers that were mounted
to ramps, that allowed an object such as a sled to be ridden down. The passengers in the
sled then coasted along the rollers down the ramps, hence the term ’roller coaster’ [1].
However, despite giving these hugely popular machines their name today, these simple
American rides had no other impact on the future of roller coasters.
The first full circuit roller coaster, Promenades Aeriennes (Aerial Walk), was built
early in the nineteenth century (1817) in France (see Figure 1) [1]. Since this achieve-
ment, designers have been utilising many different methods in order to create taller,
faster and more exciting roller coasters. Today, roller coasters come in a vast array of
shapes and sizes. Ranging from a variety of track materials, to different seat configu-
rations (e.g. inverted coasters - riders are suspended underneath the track), designers
look to create the most thrilling experience whilst keeping the passengers safe.
Figure 1: Promenades Aeriennes - first full circuit roller coaster [10].
Throughout the design phase, many complications will arise due to the high number
of forces that act on the train, as it makes its way around the circuit. However, before
designers can begin to tackle these problems, the method in which the train gathers
enough speed to complete the circuit must be determined. There are two main ways in
which this is achieved:
2
I Lift Hill - The train is pulled up a hill at the beginning of the ride. At the top of
the hill, the train is released and runs freely around the track, with only the force
due to gravity accelerating it to the required speed to complete the circuit.
II Launch System - Many roller coasters now utilise a launch system, which accelerates
the train to its top speed in a matter of seconds, so that the circuit can be completed.
Once the method of accelerating the train has been determined, designers can then
begin to work on the rest of the track layout. There are many factors that influence
the layout of a roller coaster, all of which must be taken into consideration. One of the
main factors is g-force.
G-force creates a problem for designers, because humans can lose consciousness when
subjected to a g-force greater than 5g. When a roller coaster is designed, the track layout
must compensate for this and should be designed in a specific way, so that riders are
subjected to no more than this value. This puts severe restrictions on how a track can
be laid out. Early experimentations with vertical loops put riders under as much as 12g
[1]. This caused serious injuries, identifying that the shape of the vertical loop needed
adjusting.
As the train is set into motion, multiple forces act upon it. The force due to gravity
powers the roller coaster for some, if not all of its journey around the track. There will
also be a reaction force from the track acting on the train. The majority of modern
roller coaster trains are designed with special wheel configurations, so that they have
wheels on top and underneath tubular tracks [1]. This ensures that the reaction force
is never zero, indicating that the train and track always remain in contact with each
other.
Next, we have an opposing force called the contact force, or friction. This contact
force opposes the motion of the train as it moves along the track, consequently slowing
the train down by the means of energy conversion; converting kinetic energy to heat
energy. Care should be taken to ensure that the contact force is not too great, otherwise
significant damage to the track may occur.
Drag, D, is also a very important factor to consider when designing a roller coaster.
Drag is caused by a body moving through a medium (in this case, air) and pushing
the medium out of its path. By Newton’s third law, the medium then exerts equal and
opposite forces on the body, consequently slowing the body down.
Given that roller coasters often travel at very high velocities, we consider the drag
3
equation for an object with high velocity, given by [6]
D = −µ|u|u ,
where µ is the drag coefficient and u is the velocity of the object, with magnitude, u.
The drag coefficient is determined by experiment and is affected by the shape of the
object moving through the medium. It should also be noted that |D| = −µu2 , showing
that the magnitude of the drag force is proportional to the magnitude of the velocity
squared.
In order to minimise the drag coefficient, fewer and weaker collisions between the
air and the car must occur. To achieve this, the front car of the roller coaster train,
being the biggest contributor to drag, is designed in particular ways, so that the num-
ber/magnitude of the collisions that occur is reduced. The front car usually has a small
surface area and is conical in shape, so that fewer air particles collide with the train and
the angle at which the collisions occur is reduced, making the collisions weaker.
Figure 2: Standard design for the front car of a roller coaster train.
All of these forces that act on the train makes modelling a roller coasters motion
incredibly complex. It is therefore logical to begin by making some simplifications to
the system, before we attempt to perform any calculations.
Firstly, we begin by restricting the motion to the x-z plane, with the x-axis being
the horizontal and the z-axis being the vertical. At this point we will also ignore the
effects of drag and friction. Eventually, we will introduce the effects of drag into the
problem, before including the y-axis, so that motion can be studied in three dimensions.
The second simplification is that we first model the train as a point mass positioned
on top of the track. This prevents dealing with multiple forces that vary at different
points throughout a long train at specific times. Also the shape of each car, in particular
the front car, will affect the drag, which increases the complexity of the problem. Pro-
gressing from a point mass, two point masses can be connected via a light inextensible
rod, before adding further masses to the train, so that formulae governing the motion
for an n-mass train, where n ∈ N, can be determined.
4
Once these simplifications have been made, it is possible to begin considering the
motion of a point mass around sections of track; in particular the vertical loop.
The vertical loop is one of the most simplistic inversions that can be found on a
roller coaster, although the mathematics behind it isn’t. The first vertical loop to be
constructed was in Paris in 1846 (Grand Centrifugal Railway) [1]. The ride relied on
centripetal forces to keep the car and track in contact with each other. The train would
enter the loop at high speeds, ensuring that the reaction force from the track on the
train remained positive, when fully inverted at the loop’s highest point.
Early vertical loops, including the Grand Centrifugal Railway, were designed so that
they were a perfect circle. Beginning in the x-z plane, we will consider a point mass
on a circular track and attempt to calculate the magnitude of the initial velocity that
must be given to the mass, so that it passes the highest point of the track (the top of
the vertical loop).
To find a solution to this problem, we need to consider constrained motion/systems.
As the point mass is set into motion, we will consider the effects of different forces that
are acting on it. In the most simplified case, this is the force due to gravity, along with
the normal reaction force. By applying Newton’s second law of motion, we can obtain
an equation to calculate the reaction force, along with an equation governing the motion
of the point mass along the track. This can be solved using a variety of techniques to
find an equation to calculate the magnitude of the initial velocity that is required for
the point mass to reach the top of the loop. The equation governing the reaction force
is only useful for roller coasters with particular wheel configurations and will not be
investigated further. As we progress and add the drag force into the calculations, the
equation of motion will be adjusted to account for this extra force.
By studying this original problem in more depth, we will discover that this perfect
circular shape used in original vertical loop designs, meant that the g-force experienced
by the riders was far too great, often resulting in injury. These incredibly high g-forces
were caused by the sudden change in the curvature during the transition between the
horizontal and circular track. This transition caused a massive onset of centripetal ac-
celeration, resulting in the high g-force that is experienced. By finding an equation
for g-force and then calculating the turning points of this equation, we will show when
g-force takes its maximum value for the circular loop, along with calculating this max-
imum value to prove that it is too large. It is because of these high g-forces that were
experienced, that adjustments to the shape of a vertical loop were required.
The shape of the vertical loop was altered so that it delivered a smooth, comfortable
5
ride in 1975 by Werner Stengel [1]. Mathematically, this perfect vertical loop is known
as a clothoid loop, similar to the shape of an inverted teardrop and can be modelled
by a parametric plot of two integrals, known as the Fresnel integrals. A clothoid loop’s
radius of curvature decreases as the train moves from the bottom, to the top of the loop.
Along with calculating the magnitude of initial velocity that is required for a point mass
to reach the top of a clothoid loop, it will be shown that the transition from a larger
radius of curvature at the bottom of the loop, to a smaller radius of curvature at the top,
reduces the g-force experienced by the rider to a tolerable amount. For this reason, the
clothoid loop became the preferred shape of a vertical loop for roller coaster designers
and is still used today in multiple designs.
2 Circular Loop
We begin by studying the motion of a point mass around an old fashioned, circular
loop. First, consider a point mass, m, which is attached to a circular track of radius
a, in the x-z plane (see Figure 3). We will explore two different ways of calculating
the magnitude of the mass’ initial velocity, ξ0, that is required for the mass to pass the
highest point of the track; the top of the vertical loop. Initially, the only external forces
which will be considered are the force due to gravity and the reaction force. We will then
extend the problem further by introducing an extra external force, before investigating
the problems of using a vertical loop of this shape.
First, we define the variables that will be used in the problem. We set θ as the
angle between the negative z-axis and the radial unit vector ˆr, R as the reaction force
which acts in the −ˆr-direction with magnitude R, ˆθ as the transverse unit vector, g
the acceleration due to gravity and ˆi and ˆk as the unit vectors in the x- and z-direction
respectively. Here we note that g = −gˆk, where g = 9.81ms−2.
2.1 Motion in Polar Coordinates
The problem in question considers a point mass being restricted to moving in a circle.
Therefore, we first look to solve the problem using polar coordinates. We define the
mass’ position vector as r, with dr
dt and d2r
dt2 representing the velocity and acceleration
respectively. Initially, the mass is positioned at θ = 0, at time t = 0. We want to
calculate the magnitude of the initial velocity that is required, for the mass to pass the
point θ = π. We take the radius of the circle to be r = a.
We apply Newton’s second law to the mass and look for a solution by substituting
6
Figure 3: Problem setup using polar coordinates.
into this equation. From classical mechanics, we know that the velocity of a point mass
in polar coordinates is given by
dr
dt
=
dr
dt
ˆr + r
dθ
dt
ˆθ , (2.1)
and the acceleration is given by
d2r
dt2
=
dr
dt
− r
dθ
dt
2
ˆr + r
d2θ
dt2
+ 2
dr
dt
dθ
dt
ˆθ , (2.2)
where r(t) and θ(t) are functions of time, t.
Immediately, we can simplify (2.1) and (2.2). Since r is constant, dr
dt = d2r
dt2 = 0.
Therefore, (2.1) reduces to
dr
dt
= a
dθ
dt
ˆθ , (2.3)
telling us that the velocity of the mass acts in the ˆθ-direction, tangential to the circle,
with magnitude adθ
dt . Similarly, from (2.2), we have
d2r
dt2
= −a
dθ
dt
2
ˆr + a
d2θ
dt2
ˆθ . (2.4)
Applying Newton’s second law, from (2.4) we have
F = ma −
dθ
dt
2
ˆr +
d2θ
dt2
ˆθ , (2.5)
where m is the mass and F is the total force acting on the mass.
7
Next we sum the forces that are acting on the mass to calculate F . This requires us
to calculate the component of each force acting in the ˆr- and ˆθ-direction. Resolving in
the direction of each unit vector, we find that
F = (mg cos θ − R)ˆr − (mg sin θ)ˆθ . (2.6)
Here we note that the magnitude of the reaction force in the ˆθ-direction is zero, since
R and ˆθ are orthogonal. Then, by substituting (2.6) into (2.5), we obtain
ma −
dθ
dt
2
ˆr +
d2θ
dt2
ˆθ = (mg cos θ − R)ˆr − (mg sin θ)ˆθ . (2.7)
By taking the dot product of this equation with both ˆr and ˆθ, two scalar equations are
obtained. We find that
ˆr · (2.7) : −ma
dθ
dt
2
= mg cos θ − R , (2.8)
ˆθ · (2.7) : ma
d2θ
dt2
= −mg sin θ . (2.9)
Equation (2.8) can be rearranged so that the reaction force can be determined through-
out the mass’ motion. Equation (2.9) is a second order differential equation, which can
be solved to find an equation calculating the magnitude of the mass’ velocity.
We attempt to solve (2.9) by defining the new variable, ξ = dθ
dt , so that differentiating
using the chain rule gives
d2θ
dt2
=
d
dt
dθ
dt
=
d
dθ
dθ
dt
dθ
dt
=
dξ
dθ
ξ .
By substituting this into (2.9), we obtain a separable equation, such that
aξ
dξ
dθ
= −g sin θ . (2.10)
Separating the variables and integrating, gives
1
2
ξ2
=
g
a
cos θ + B , (2.11)
where B is a constant of integration. To find this constant of integration, an initial
condition is required. At t = 0, the mass is at θ = 0. Therefore, at t = 0, i = ˆθ. Hence,
given a velocity, u, with initial magnitude ξ0, from (2.3) we have
dr
dt
= u = ξ0i = a
dθ
dt
ˆθ = aξ ˆθ ,
8
at t = 0. Therefore, the initial condition is given by θ = 0 and ξ = ξ0
a , at t = 0. This
condition will be used throughout Section 2. We apply this condition to equation (2.11)
and calculate B, so that after some slight simplification, we arrive at
1
2
ξ2
=
g
a
(cos θ − 1) +
ξ2
0
2a2
. (2.12)
To calculate ξ0 so that the mass passes the highest point of the loop (θ = π), we
introduce a final condition, such that ξ = 0, when θ = π, which will be used throughout
Section 2. Applying this condition to equation (2.12) gives an equation that determines
a critical value for the magnitude of the initial velocity that should be given to the mass,
for it to just reach the top of the loop. Should the mass be given an initial velocity with
a higher magnitude than this critical value, the mass will pass the highest point of the
loop.
After applying the final condition, equation (2.12) reduces to
ξ2
0
2a2
=
2g
a
,
which can be rearranged so that we have
ξ0 = 2
√
ag . (2.13)
Therefore, for a given radius, a, and initial velocity magnitude, ξ0, we have
ξ0 > 2
√
ag mass passes θ = π
ξ0 < 2
√
ag mass doesn’t pass θ = π
(2.14)
We can perform a rough check of these inequalities, by calculating ξ0 for a real life
example. The tallest vertical loop in the world is the main element of the roller coaster
Full Throttle, at Six Flags Magic Mountain, California (see Figure 4) [7]. This roller
coaster utilises a launch system, allowing the vertical loop to be the tallest part of the
ride. Due to this, the top speed of the roller coaster occurs approximately at the point
when the train enters the loop, at θ = 0, to make sure the train successfully completes
the inversion. Although the loop is not a perfect circle, we will assume here that it is.
The loop has a diameter of 38.75m, therefore its radius is a = 19.375m [7]. Substitut-
ing this into (2.13), we find that ξ0 ≈ 61.68mph. This critical value of ξ0 is less than the
70mph top speed of the roller coaster [9]. Since the roller coaster successfully completes
the inversion, (2.14) holds, adding confidence that the inequalities are correct. When
calculating the initial velocity of a roller coaster, designers want to obtain a highly ac-
curate numerical value. Therefore, it should be noted that no rounding has taken place
until the final answer is found, so that the error in the calculations is minimised.
9
Figure 4: The tallest vertical loop in the world; the main element of the roller coaster
Full Throttle, at Six Flags Magic Mountain, California [8].
2.2 Energy Argument
It is possible to calculate (2.13) in an alternate way, by means of an energy argument.
From classical mechanics, we know that kinetic energy is given by T = 1
2m|u|2 and
the gravitational potential energy is given by V = mgh, where h is the vertical distance
between the x-axis and the mass’ position. As the problem does not involve any resisting
Figure 5: Resolving diagram for an energy argument.
forces, no energy will be lost. Therefore, the system is a conservative system. This allows
us to write
T + V = B , (2.15)
where B is a constant.
First, using equation (2.3) we note that |u|2 = a2 dθ
dt
2
, giving T = 1
2ma2 dθ
dt
2
.
Then, resolving from Figure 5, we find that a − h = a cos θ, so that V = mga(1 − cos θ).
Substituting T and V into (2.15), gives
1
2
ma2 dθ
dt
2
+ mga(1 − cos θ) = B .
10
Setting ξ = dθ
dt and applying the initial condition defined in Section 2.1, we calculate B
and arrive at exactly equation (2.12). Equation (2.13) can then be derived from this as
before.
2.3 Air Resistance/Drag
2.3.1 High Air Resistance
We now look to add the effects of drag, D, into the problem. We begin with the case
where the drag coefficient, µ, is large (µ >> 1) and derive an equation for the critical
magnitude of the initial velocity, ξ0. An alternate equation can then be found by making
adjustments to this equation, for the case when µ is small (µ << 1).
As previously discussed in Section 1, when an object has a high velocity, the drag
force is given by
D = −µ|u|u . (2.16)
From Section 2.1, equation (2.3) tells us that u = adθ
dt
ˆθ, so that |u| = adθ
dt . Therefore,
we can rewrite D as
D = −µa2 dθ
dt
2
ˆθ .
Next, we apply Newton’s second law to the mass. Starting with equation (2.7), we
add the additional drag force to give
ma −
dθ
dt
2
ˆr +
d2θ
dt2
ˆθ = (mg cos θ − R)ˆr − mg sin θ − µa2 dθ
dt
2
ˆθ .
Taking the dot product of this equation with ˆθ, we obtain
ma
d2θ
dt2
= −mg sin θ − µa2 dθ
dt
2
.
Then, by introducing the variable ξ = dθ
dt , where d2θ
dt2 = ξ dξ
dθ , this equation becomes
maξ
dξ
dθ
= −mg sin θ − µa2
ξ2
. (2.17)
The −µa2ξ2 term makes this equation non-linear. To make this equation linear, we
define a new variable w = ξ2, so that differentiating w with respect to θ using the chain
rule gives
dw
dθ
=
dw
dξ
dξ
dθ
= 2ξ
dξ
dθ
. (2.18)
Substituting w and (2.18) into (2.17) and rearranging, we then arrive at
dw
dθ
+
2µaw
m
= −
2g
a
sin θ . (2.19)
11
This equation is of the form dw
dθ + p(θ)w = q(θ), therefore, it can be solved by using
an integrating factor. After calculating the integrating factor and multiplying (2.19) by
this factor, we may write
d
dθ
[we
2µa
m
θ
] = −
2g
a
sin θe
2µa
m
θ
.
Integrating this equation then gives
we
2µa
m
θ
= −
2g
a
I + A , (2.20)
where A is a constant of integration, which can be determined by the initial condition
and I is given by
I = sin θe
2µa
m
θ
dθ ,
which must be calculated.
To calculate I, we re-write sin θ in terms of the complex exponential. Setting α = 2µa
m ,
the integral becomes
I = sin θeαθ
dθ = Im(eiθ
)eαθ
dθ ,
where i =
√
−1 and eiθ = cos θ + i sin θ. Adding the powers of the exponentials and
integrating gives
I = Im
1
α + i
e(α+i)θ
.
Next, we multiply by α−i
α−i and take the imaginary part, so that after some simplification,
we arrive at
I =
1
α2 + 1
(α sin θ − cos θ)eαθ
.
Then, by substituting back in for α and substituting I into (2.20), we obtain
we
2µa
m
θ
=
2gm
a(4µ2a2 + m2)
(m cos θ − 2µa sin θ)e
2µa
m
θ
+ A . (2.21)
Noting that the newly introduced variable was given by w = ξ2, the initial condition
introduced in Section 2.1 becomes w(0) =
ξ2
0
a2 . Using this condition, we can calculate
the constant of integration, A, so that equation (2.21) becomes
we
2µa
m
θ
=
2gm
a(4µ2a2 + m2)
(m cos θ − 2µa sin θ)e
2µa
m
θ
+
ξ2
0
a2
−
2gm2
a(4µ2a2 + m2)
. (2.22)
We now apply the final condition introduced in Section 2.1 to this equation, so that
we can determine an equation for the critical magnitude of the initial velocity. This
12
final condition requires w = 0, at θ = π. By substituting this condition into (2.22) and
rearranging, the critical magnitude equation is given by
ξ2
0 =
2gm2a
(4µ2a2 + m2)
(1 + e
2µa
m
π
) . (2.23)
To check that this equation is consistent with the previous case where drag was neglected,
we set µ = 0, giving exactly equation (2.13) as required.
2.3.2 Low Air Resistance
Many roller coaster trains are cleverly designed to reduce the drag coefficient as much
as possible, so that the drag has minimal effect on the magnitude of the initial velocity
that is required for the train to reach the top of the vertical loop. Therefore, we now
consider the case where we have very minimal drag (µ << 1) and find an equation for
ξ0, which can be used for a train which has been made as aerodynamic as possible.
We begin by re-writing equation (2.23) as
ξ2
0 =
2gm2a
m2(4µ2a2
m2 + 1)
(1 + e
2µa
m
π
) =
2ga
(α2 + 1)
(1 + eαπ
) , (2.24)
where α = 2µa
m . We can expand eαπ in Taylor series and (α2 + 1)−1 using a binomial
expansion, such that
eαπ
≈ 1 + απ +
α2π2
2
+ O(α3
) ,
(α2
+ 1)−1
≈ 1 − α2
+ O(α3
) .
By noting that µ << 1 and therefore α << 1, terms of order O(α2) and higher are
negligible. Hence, using these expansions, (2.24) becomes
ξ2
0 ≈ 2ga(1 + (1 + απ + O(α2
)))(1 − O(α2
)) .
Multiplying out the brackets and substituting back in for α gives
ξ0 = 2
√
ga 1 +
µaπ
m
1
2
,
where we again use a binomial expansion, neglecting terms of order O(µ2), to give
ξ0 ≈ 2
√
ga 1 +
µaπ
2m
= 2
√
ga +
µa
3
2 πg
1
2
m
. (2.25)
Clearly if µ = 0, this equation for ξ0 is the same as equation (2.13), as required.
For a roller coaster train, particularly when loaded with passengers, m will be quite
large. By also noting that µ << 1, it can be seen that the extra term appearing in the
equation for ξ0 given by (2.25), will have minimal effect on the value of ξ0, showing the
benefit of a roller coaster train being aerodynamically designed.
13
2.4 G-Force
After calculating an initial velocity for a roller coaster, designers are then faced with
another obstacle known as g-force. The g-force acting on an object is defined as the ratio
between the force developed from the object accelerating by a cause other than gravity,
relative to the force due to gravity on the object [1]. Designers often use g-force to
their advantage, by creating moments of weightlessness, along with moments of extreme
heaviness for a rider. This is achieved by using specific track shapes, to continually vary
the g-force that a rider experiences, throughout the course of the roller coaster.
However, despite g-forces creating multiple thrilling sensations for a roller coaster’s
riders, they are the biggest limiting factor on the production of more extreme roller
coasters. Should riders experience greater than +5g, or less than −5g, serious injury
may occur, something designers clearly want to avoid [1]. It is for this reason why the
use of a circular loop in a roller coaster’s layout was stopped and a safer, alternative
shape was used. By deriving an equation for the g-force, we can identify the problems
of using a vertical loop of this shape and calculate where these problems occur. For
simplicity, air resistance and friction will be neglected throughout.
It is sufficient to show that a circular loop is unsafe by showing that the magnitude
of g-force exceeds 5g, regardless of direction. Directly from the definition of g-force, we
can write an equation for the magnitude of g-force, G, that is experienced by the mass,
at any point on the loop. Recalling the definition, we have that
G =
m d2r
dt2
m|g|
=
d2r
dt2
|g|
. (2.26)
Immediately it can be seen that g-force is independent of mass. This identifies that two
riders of different mass will experience the same g-force as each other.
Using equation (2.4), we can write d2r
dt2 as
d2r
dt2
= a
dθ
dt
4
+
d2θ
dt2
2
1
2
. (2.27)
Recalling that ξ = dθ
dt , so that d2θ
dt2 = ξ dξ
dθ , we can make substitutions for dθ
dt and d2θ
dt2 using
equations (2.10) and (2.12) respectively. Hence, by making these substitutions and then
substituting (2.27) into (2.26), we obtain
G =
a
g
g2
a2
sin2
θ +
ξ2
0
a2
−
2g
a
+
2g
a
cos θ
2
1
2
. (2.28)
14
Solving dG
dθ = 0 will give the turning points of equation (2.28). One of these turning
points will be the value of θ at which the magnitude of g-force takes its maximum value.
Given that we are only concerned with motion for 0 ≤ θ ≤ π, we will neglect any turning
points that are not within this interval. Furthermore, by calculating d2G
dθ2 , we can classify
the turning points, which in turn allows us to identify what part of the loop generates
this maximum value, along with calculating an exact, maximum value of G.
We begin by squaring equation (2.28), which, after simplification, gives an equation
which can be differentiated much more easily, term-by-term. Differentiating using the
chain rule, after further simplification, we arrive at
2G
dG
dθ
= −6 sin θ cos θ −
4ξ2
0
ag
sin θ + 8 sin θ . (2.29)
For turning points, dG
dθ = 0. Setting dG
dθ = 0, equation (2.29) simplifies to
sin θ −6 cos θ −
4ξ2
0
ag
+ 8 = 0 .
To satisfy this equation, either sin θ = 0, or −6 cos θ −
4ξ2
0
ag + 8 = 0. Using equation
(2.13), we make the substitution ξ0 = 2
√
ag, whereby it can be shown that the latter of
these equalities requires | cos θ| > 1. This clearly cannot be true. Therefore, the turning
points of equation (2.28), for 0 ≤ θ ≤ π, are θ = 0 and θ = π.
In order to classify these turning points, we must differentiate equation (2.29) with
respect to θ using the product rule, so that we obtain
2G
d2G
dθ2
+ 2
dG
dθ
2
= 6 sin2
θ − 6 cos2
θ −
4ξ2
0
ag
cos θ + 8 cos θ . (2.30)
By definition, when d2G
dθ2 > 0 we have a minimum turning point and when d2G
dθ2 < 0 we
have a maximum turning point. Noting that G > 0 by definition, we substitute θ = 0
and θ = π into equation (2.30) and use equation (2.13) to determine that
θ = 0, dG
dθ = 0 2Gd2G
dθ2 = 2 −
4ξ2
0
ag < 0 → maximum
θ = π, dG
dθ = 0 2Gd2G
dθ2 =
4ξ2
0
ag − 2 > 0 → minimum
Given that θ = 0 is the only maximum turning point, the magnitude of g-force is
greatest at θ = 0. This is the point of transition between the horizontal and circular
track, otherwise known as the entrance to the loop.
Since the second half of the vertical loop can be generated by reflection, the g-force
for the second half of the loop can also be generated in the same way. This implies that
for the entire vertical loop (0 ≤ θ ≤ 2π), we have another maximum turning point at
15
θ = 2π; the exit of the vertical loop. When substituted into equation (2.28), θ = 0 and
θ = 2π give an equivalent maximum value of G, denoted Gmax.
We can calculate Gmax for any circular vertical loop, by substituting θ = 0 (or
θ = 2π) into equation (2.28). We reconsider the vertical loop from the Full Throttle
roller coaster, described in Section 2.1. Substituting a = 19.375m and ξ0 = 70mph
into equation (2.28), we find that Gmax ≈ 5.15g. Since Gmax > 5g, Gmax is too large,
showing that this circular shape is not an acceptable shape for a vertical loop.
The reason for large g-forces being experienced at the entrance/exit of a circular
loop, is due to the large, sudden change in curvature at the point of transition be-
tween the horizontal and circular track. This indicates that to make the vertical loop
a more comfortable experience for riders, the current large change in curvature at the
entrance/exit of the loop must be decreased. This will reduce the g-force experienced,
so that for the entirety of the loop, the g-force remains below 5g. This can be achieved
by using properties of the clothoid curve.
3 Clothoid Loop
The clothoid curve, otherwise known as the Euler/Cornu spiral, has been utilised by
roller coaster designers to avoid exerting high g-forces on riders. It has the property
that its radius of curvature is inversely proportional to the distance from the center
of the spiral, which is equivalent to the curvature varying linearly with arc length [1].
This linear variation property means that a vertical loop modelled in the shape of this
curve will have a very minimal change in curvature at the entrance and exit of the loop.
Consequently, this should reduce the g-force experienced to a tolerable amount, making
the roller coaster safer and more enjoyable for the rider.
3.1 Defining the Curve
Before we can consider the motion of a mass around a two-dimensional clothoid loop,
we must first find a system of equations that allow us to compute the clothoid curve.
This requires some simple definitions, which can then be manipulated to generate the
curve.
We begin by considering a general curve, C, with arc length, s, in the x-z plane (see
Figure 6). A position on this curve is represented by the vector r, where r is represented
by two parametric equations, x(s) and z(s), such that r = (x(s), z(s)). We use ˆn and ˆt
as the unit normal and unit tangent vectors respectively.
16
Figure 6: General curve, C, with defined position and unit vectors.
We now define ˆt and ˆn in terms of x(s) and z(s), along with the curvature, κ. At
any point on C, the tangential unit vector is given by
ˆt =
dr
ds
=
dx
ds
ˆi +
dz
ds
ˆk , (3.1)
where ˆi and ˆk are unit vectors in the x- and z-directions respectively. Using the orthog-
onal property (ˆt · ˆn = 0), we then define ˆn as
ˆn = −
dz
ds
ˆi +
dx
ds
ˆk . (3.2)
Finally, the curvature is defined as [4]
κ =
dˆt
ds
· ˆn . (3.3)
Defining θ as the angle between ˆt and the horizontal x-axis, equation (3.3) can then
be rewritten as a relation between θ and the arc length, s. Beginning by taking the dot
product of ˆt with ˆi and ˆk, we find from the definition of the dot product that ˆi·ˆt = cos θ
and ˆk · ˆt = sin θ. Therefore, from equations (3.1) and (3.2), we can deduce that
dx
ds
= cos θ ,
dz
ds
= sin θ , (3.4)
ˆt = cos θˆi + sin θˆk , (3.5)
ˆn = − sin θˆi + cos θˆk . (3.6)
Next we differentiate ˆt with respect to s using the chain rule and note that dˆt
dθ = ˆn.
This gives
dˆt
ds
=
dˆt
dθ
dθ
ds
= ˆn
dθ
ds
. (3.7)
17
Substituting (3.7) into (3.3), we obtain
κ = ˆn ·
dˆt
ds
= ˆn · ˆn
dθ
ds
=
dθ
ds
. (3.8)
From (3.4) and (3.8), we now have a system of ordinary differential equations. This
system can be written as
d
ds




x
z
θ



 =




cos θ
sin θ
κ



 . (3.9)
Solving this system will produce two parametric equations, x(s) and z(s), which describe
the clothoid curve.
To begin solving (3.9), we recall the definition of the clothoid curve. Since the
clothoid’s curvature varies linearly with arc length, we can write κ(s) = αs, where α is
a constant. Taking the third equation in system (3.9) and substituting in κ = αs, we
use separation of variables to find an equation for θ, such that
θ =
αs2
2
+ θ0 ,
where θ0 is a constant of integration. Without loss of generality, we take θ0 = 0, so that
θ = αs2
2 .
We can now substitute θ into the first two equations in system (3.9), before separating
the variables and integrating. To integrate the right hand side, we introduce the dummy
variable ˆs and integrate with respect to ˆs between zero and s, to give
x =
s
0
cos
αˆs2
2
dˆs , z =
s
0
sin
αˆs2
2
dˆs . (3.10)
Then, by letting ˆs =
√
π√
α
t, so that dˆs =
√
π√
α
dt, we obtain
x =
√
π
√
α
√
α
√
π
s
0
cos
π
2
t2
dt , (3.11)
z =
√
π
√
α
√
α
√
π
s
0
sin
π
2
t2
dt . (3.12)
These two integrals are otherwise known as the Fresnel integrals [2]. A parametric plot
of these two integrals will produce the clothoid curve required.
The Fresnel integrals can also be written as two power series expansions. These
expansions converge rapidly for all s (not shown here). For a given value of α, the two
series expansions are given by
x =
√
π
√
α
∞
n=0
(−1)n
√
α√
π
s
4n+1
(π
2 )2n
(2n)!(4n + 1)
,
18
z =
√
π
√
α
∞
n=0
(−1)n
√
α√
π
s
4n+3
(π
2 )2n+1
(2n + 1)!(4n + 3)
,
which can be found in [2]. These can be used to plot the clothoid curve on Matlab.
For a vertical loop, the entire clothoid curve is not required. It is important when
generating the curve that only the first half of a vertical loop is produced, instead of
allowing the curve to become a spiral. The second half of the loop can then be generated
by reflection. To ensure that only the first half of a vertical loop is produced, we require
a final condition.
To find this final condition, we recall the z-integral from (3.10), before the change of
variables took place. Differentiating z with respect to ˆs and setting dz
dˆs = 0, we calculate
the first value of ˆs (excluding ˆs = 0), which satisfies this equation. This value represents
the arc length at the point when the clothoid curve becomes horizontal for the first
time; the top of the vertical loop. This arc length will be denoted s∗. Calculating dz
dˆs
and setting dz
dˆs = 0, we find that
dz
dˆs
= sin
αˆs2
2
= 0 ,
which first occurs when αˆs2
2 = π. Hence, s∗ =
√
2π√
α
. This value will be used as part of
the final condition in all calculations throughout this section.
Using this condition, it is now possible to compute a variety of vertical loops for
different values of α (see Figure 7). By producing a number of curves of varying size,
Figure 7: Left: positive clothoid curve. Right: variety of half clothoid loops for different
values of α.
it can be seen there is a strong relation between every clothoid curve as the value of
19
α changes. As shown in Figure 7, the end point of each curve can be connected by a
straight line passing through the origin.
We can verify this result by substituting s = s∗ =
√
2π√
α
into (3.10), giving us two
integrals defined as
X =
√
π
√
α
√
2
0
cos
π
2
t2
dt , Z =
√
π
√
α
√
2
0
sin
π
2
t2
dt .
Then, by defining a shorthand notation for the Fresnel integrals such that
C(s) =
s
0
cos
πt2
2
dt , S(s) =
s
0
sin
πt2
2
dt ,
we obtain
Z
X
=
√
π√
α
S(
√
2)
√
π√
α
C(
√
2)
=
S(
√
2)
C(
√
2)
,
which is independent of α, thus verifying the result.
3.2 Motion Along a General Curve
Using the Fresnel integrals (given by (3.11) and (3.12)) in conjunction with the final
condition for arc length, we can now compute half of a clothoid loop of any height. To
allow us to calculate the magnitude of the initial velocity that should be given to a point
mass so that it reaches the top of a clothoid loop, we require an equation governing the
motion of a mass around the loop.
Considering the setup from Figure 6, we will apply Newton’s second law to a point
mass and look to find an equation of motion for a mass along a general curve, C. Once
calculated, equations describing a specific curve, in this case the clothoid loop, can be
applied to this equation of motion, so that we can study the motion of a mass around
the clothoid loop.
To substitute into Newton’s second law, we must sum the total forces acting on the
mass and calculate a formula for d2r
dt2 ; the acceleration of the mass. First we calculate
d2r
dt2 . Differentiating the position vector, r, with respect to t using the chain rule gives
dr
dt
=
dr
ds
ds
dt
.
Since the velocity acts in the tangential direction, dr
ds = ˆt. Hence,
dr
dt
= ˆt
ds
dt
. (3.13)
20
Differentiating again using the product rule, we obtain
d2r
dt2
= ˆt
d2s
d2t
+
dˆt
dt
ds
dt
,
which can be re-written by use of the chain rule, such that
d2r
dt2
= ˆt
d2s
d2t
+
dˆt
ds
ds
dt
ds
dt
= ˆt
d2s
d2t
+
dˆt
ds
ds
dt
2
. (3.14)
Using equations (3.7) and (3.8), we note that dˆt
ds = ˆndθ
ds = ˆnκ. From this, we can make
further simplifications to (3.14), so that d2r
dt2 becomes
d2r
dt2
= ˆt
d2s
dt2
+ ˆnκ
ds
dt
2
. (3.15)
Next, we sum the forces acting on the mass, initially ignoring the effects of drag and
friction. Since the reaction force, R, always acts in the ˆn-direction and the force due to
gravity acts in the −ˆk-direction, the total force acting on the mass, F , is given by
F = Rˆn − mgˆk .
Hence, Newton’s second law gives
Rˆn − mgˆk = m ˆt
d2s
dt2
+ ˆnκ
ds
dt
2
. (3.16)
Similarly to the circular loop example in Section 2.1, we want to consider the ˆn and
ˆt components of this equation. Therefore, by taking the dot product of (3.16) with ˆn
and ˆt, we obtain
ˆn · (3.16) : R − mgˆk · ˆn = mκ
ds
dt
2
, (3.17)
ˆt · (3.16) : −mgˆk · ˆt = m
d2s
dt2
. (3.18)
Recalling the definitions of ˆt and ˆn from (3.5) and (3.6) respectively, we substitute into
equations (3.17) and (3.18), so that after rearrangement, (3.17) gives an equation for
the reaction force, which is given by
R = mκ
ds
dt
2
+ mg cos θ , (3.19)
whilst (3.18) gives us a second order, ordinary differential equation, such that
d2s
dt2
+ g sin θ = 0 . (3.20)
By defining θ in terms of arc length for a specific curve, equation (3.20) can now be
solved to obtain an equation for the magnitude of the mass’ velocity. It should be noted
that by redefining θ in Section 2, we could also solve equation (3.20) for the circular
loop, to give us exactly equation (2.13), as before.
21
3.3 Motion Around the Clothoid Loop
We now have an equation of motion for a mass along a general curve, given by (3.20).
By applying equations used to describe the clothoid curve to this equation of motion,
it is possible to calculate an equation which determines the magnitude of the initial
velocity that is required for a point mass to reach the top of a loop, as done previously
for the circular vertical loop in Section 2.1.
Equation (3.20) is initially an equation of three variables. In Section 3.1, we found
that θ = αs2
2 for the clothoid. Substituting this into (3.20) gives
d2s
dt2
+ g sin
αs2
2
= 0 , (3.21)
so that we now have a differential equation in terms of only two variables, s and t.
Similarly to the approach in Section 2.1, we introduce the variable ξ = ds
dt , so that via
the chain rule, d2s
dt2 = ξ dξ
ds . Substituting this into equation (3.21), we obtain a separable
equation, which is given by
ξ
dξ
ds
+ g sin
αs2
2
= 0 .
Introducing the dummy variable ˆs, we separate the variables and integrate between zero
and s in anticipation of the initial condition, ξ(0) = ξ0, where ξ0 is the magnitude of
the initial velocity. This gives
1
2
ξ2
= −g
s
0
sin
αˆs2
2
dˆs +
1
2
ξ2
0 . (3.22)
The integral in this equation can be rewritten by setting ˆs =
√
π√
α
t, so that dˆs =
√
π√
α
dt,
as seen in Section 3.1. By making this change of variables, equation (3.22) becomes
1
2
ξ2
= −g
√
π
√
α
√
α
√
π
s
0
sin
π
2
t2
dt +
1
2
ξ2
0 . (3.23)
Recalling the shorthand notation for the Fresnel integrals, we can rewrite equation
(3.23), such that
1
2
ξ2
= −g
√
π
√
α
S
√
α
√
π
s +
1
2
ξ2
0 . (3.24)
To obtain an equation for the critical value of ξ0, a final condition is required. We
previously calculated in Section 3.1 that at the top of the clothoid loop, s = s∗ =
√
2π√
α
.
Therefore, the final condition for the mass to just reach the top of the loop is given by
ξ(s∗) = 0. Applying this final condition to equation (3.24) gives
ξ2
0 = 2g
√
π
√
α
S(
√
2) . (3.25)
22
Here we note that S(
√
2) is a known value and can be calculated by the use of Maple.
Equation (3.25) can also be derived by using an energy argument. Given that |u| =
|ˆtds
dt | = ds
dt , since |ˆt| = 1 and ds
dt > 0, the kinetic energy of the system is given by
T = 1
2m ds
dt
2
. Then, since the system is conservative, by substituting the z-Fresnel
integral given by (3.12) into the equation for gravitational potential energy, we can
write
1
2
m
ds
dt
2
+ mg
√
π
√
α
S
√
α
√
π
s = B ,
where B is a constant. Reintroducing the variable ξ = ds
dt and dividing by m gives
1
2
ξ2
+ g
√
π
√
α
S
√
α
√
π
s = D ,
where D is a new constant. By applying the initial condition it can then be shown that
D = 1
2ξ2
0. Finally, after applying the final condition to this equation and rearranging,
we arrive at equation (3.25), as required.
For a given α, it is now possible to determine the minimum magnitude of the initial
velocity that must be given to the mass, so that it reaches the top of the vertical loop.
By specifying a height, h, for the vertical loop, we substitute s = s∗ and z = h into the
z-Fresnel integral given by (3.12), so that we can rearrange to give the required value of
α for this height. This can then be used in equation (3.25) to calculate ξ0.
It is useful to compare the required initial velocity for a clothoid loop with a circular
loop of identical height, so that it can be assessed how using a different loop shape
affects the magnitude of the initial velocity that is required.
We recall the Full Throttle example used in Section 2.1. The height of the vertical
loop is 38.75m, which occurs when s = s∗. By setting z = 38.75m, evaluating the
z-Fresnel integral at s = s∗ using Maple and rearranging, we find that α ≈ 1.07 × 10−3.
Here we note that this truncated value of α is not used to calculate ξ0. Instead, the exact
value is used to minimise the error in the calculation. Substituting the exact values into
equation (3.25), gives a critical magnitude of ξ0 ≈ 61.68mph, which is identical to the
value calculated for the circular loop. To understand why these values are identical, a
general energy argument can be formulated, showing that the equation calculating the
magnitude of the initial velocity of a mass along a general curve is independent of curve
shape.
3.4 General Energy Argument
Consider a general curve with arc length s, connecting two fixed points, P and Q, in
the x-z plane (see Figure 8). P is at height z = a and Q is at height z = a + H, where
23
H is the vertical distance between P and Q and a ∈ R. Neglecting air resistance and
friction, we use simple energy definitions to find an equation to calculate the magnitude
of the initial velocity, ξ0, that is required to launch a point mass from P to Q, so that
the mass’ velocity is zero as it reaches point Q.
Figure 8: General curve between two points, P and Q.
Since all resisting forces are neglected, the system is conservative. Hence, we can
write T + V = B, where B is constant. In Section 3.3, we found that T = 1
2m ds
dt
2
.
Thus, noting that V = mgz, we have
1
2
m
ds
dt
2
+ mgz = B .
Defining ξ0 as the magnitude of the initial velocity, the initial condition requires that
ds
dt = ξ0, at z = a. This implies that B = 1
2mξ2
0 +mga. Next we apply the final condition,
which requires ds
dt = 0, at z = a + H. After some simple rearranging, we are left with
ξ2
0 = 2gH , (3.26)
which is clearly independent of the path taken as required.
This result proves that when neglecting air resistance and friction, for a given height
of vertical loop, the shape of the vertical loop has no effect on the value of ξ0. Pro-
viding the system is conservative, for loop shapes which are complicated to describe
mathematically, equation (3.26) may be used as a useful shortcut to avoid complicated
calculations when determining the value of ξ0.
3.5 G-Force
Since the clothoid curve’s properties are used primarily to reduce the g-force experienced
by riders, it is vital that the g-force experienced when travelling around a clothoid loop
24
is calculated, to verify that riders are indeed safe. By recalling the definition for the
magnitude of g-force, G, we will apply equations from Section 3.3 to the definition to
acquire an equation for G, in terms of arc length, s. Air resistance and friction are
neglected throughout.
We begin by substituting equation (3.15) into the equation for the magnitude of
g-force, which is given by (2.26), so that we have
G =
1
g
d2s
dt2
ˆt + κ
ds
dt
2
ˆn . (3.27)
A substitution can then be made for d2s
dt2 using equation (3.20). Similarly, by noting that
ds
dt = ξ, we can also make a substitution for ds
dt
2
using equation (3.24). Therefore, by
substituting equations (3.20) and (3.24) into equation (3.27) and acknowledging that for
the clothoid, θ = αs2
2 and κ = αs, we obtain
G =
1
g
−g sin
αs2
2
ˆt + αs −2g
√
π
√
α
S
√
α
√
π
s + ξ2
0 ˆn .
This is equivalent to writing
G =
1
g
g2
sin2 αs2
2
+ α2
s2
−2g
√
π
√
α
S
√
α
√
π
s + ξ2
0
2
1
2
. (3.28)
Solving the equation dG
ds = 0 gives the turning points of equation (3.28). Further-
more, by calculating d2G
ds2 , we can classify these turning points as either minimum or
maximum turning points. One of these maximum turning points is the value of s at
which the magnitude of g-force is maximum. This maximum magnitude is denoted by
Gmax. By substituting each maximum turning point into equation (3.28), the maximum
turning point which calculates Gmax can be determined, along with a numerical value
for Gmax.
For the first half of the clothoid loop, 0 ≤ s ≤ s∗. Therefore, turning points that
are not within this interval are neglected. Since the upper bound of the interval is
dependent on α, we must first specify a value of α before the turning points can be
calculated. It is useful to compare the magnitude of the g-force that is experienced from
using a clothoid loop, with a circular loop of identical height. Therefore, we recall the
vertical loop from the Full Throttle roller coaster introduced in Section 2.1, for which
we previously calculated Gmax ≈ 5.15g for the circular shape loop in Section 2.4. In
Section 3.3, it was calculated that α ≈ 1.07 × 10−3 for this vertical loop, which will
now be used in the interval, so that the turning points of equation (3.28) can be can be
calculated.
25
Given the complexity of equation (3.28), we use Maple to calculate the turning
points of the equation. By solving dG
ds = 0 on Maple, we find that equation (3.28) has
three turning points in the range 0 ≤ s ≤ s∗. These are given by s = 0, s = s∗ and
s ≈ 40.76m, where s ≈ 40.76m is just over halfway along the first half of the clothoid
loop. Furthermore, by calculating d2G
ds2 and substituting in the turning points, we find
that s = 0 and s = s∗ are minimum turning points and s ≈ 40.76m is a maximum
turning point. Since s ≈ 40.76m is the only maximum turning point of equation (3.28),
Gmax occurs at this value of s.
Recalling that ξ0 = 70mph for the Full Throttle roller coaster, we substitute the
non-truncated values of s ≈ 40.76m, α ≈ 1.07 × 10−3 and ξ0 = 70mph into equation
(3.28), giving Gmax ≈ 2.50g. Clearly Gmax is substantially lower than 5g, showing that
the clothoid is an acceptable shape for a vertical loop.
For the clothoid loop, Gmax is less than half the value of Gmax for the circular loop.
To visualise how G varies as the mass travels around the two loop shapes, we plot G
against s using equations (2.28) and (3.28) (see Figure 9), noting that s = aθ for the
circular loop.
Figure 9: Graph to show how the magnitude of g-force, G, varies with arc length, s,
for a circular loop (red) and a clothoid loop (blue) of identical height (38.75m), for
0 ≤ s ≤
√
2π√
α
.
The main problem with the circular loop is the large sudden change in curvature at
26
the point of transition between the horizontal and circular track. It can be seen from
Figure 9 that the clothoid loop has removed this problem, because G steadily increases
from zero, as opposed to the circular loop, where G decreases from its maximum value.
It is also clear that Gmax for the circular loop is approximately double Gmax for the
clothoid loop, adding confidence that the calculations of Gmax in Sections 2.4 and 3.5
are correct. It should also be noted that s∗ for the circular loop is smaller than s∗ for
the clothoid loop. Therefore, in Figure 9, the g-force for the circular loop is plotted for
more than the first half of the loop.
3.6 Air Resistance/Drag
Now that we have verified that the use of the clothoid curve to model a vertical loop
reduces the g-force experienced by riders to a tolerable amount, we look to add an extra
external force into the problem; the drag force.
The drag force that we add into the problem is given by equation (2.16), such that
D = −µ|u|u. Since the velocity, u, points in the tangential direction, we can rewrite
D in terms of ˆt. We can then find an equation of motion, so that an equation which
determines the magnitude of the initial velocity that is required for a point mass to
reach the top of a clothoid loop, with drag included, can be found.
From equation (3.13), we know that u = ˆtds
dt . Substituting this into (2.16), we have
D = −µ
ds
dt
ds
dt
ˆt = −µ
ds
dt
2
ˆt ,
since |ˆt| = 1 and ds
dt > 0.
Now that we have a relation between D and ˆt, we can substitute into Newton’s
second law, giving us equation (3.16) with an additional term on the left hand side, such
that
Rˆn − mgˆk − µ
ds
dt
2
ˆt = m ˆt
d2s
dt2
+ ˆnκ
ds
dt
2
. (3.29)
In Section 3.1, we found that θ = αs2
2 for the clothoid. Hence, by taking the dot product
of equation (3.29) with ˆt and substituting in for θ, we obtain
−mg sin
αs2
2
− µ
ds
dt
2
= m
d2s
dt2
. (3.30)
To solve this equation, we will make use of a substitution. We again introduce the
variable ξ = ds
dt , so that via the chain rule, d2s
dt2 = ξ dξ
ds . Substituting this variable into
equation (3.30) gives
mξ
dξ
ds
= −µξ2
− mg sin
αs2
2
. (3.31)
27
By making a further substitution of the form w = ξ2, so that via the chain rule, dw
ds =
2ξ dξ
ds , we can rearrange equation (3.31) to give
dw
ds
+
2µw
m
= −2g sin
αs2
2
. (3.32)
This equation is now of the correct form so that it can be solved by using an integrating
factor. Letting β = 2µ
m , we calculate the integrating factor and multiply equation (3.32)
by this factor. Then we may write
d
ds
weβs
= −2g sin
αs2
2
eβs
. (3.33)
We now integrate this equation, introducing the limits, zero and s, in anticipation of
the initial condition, w(0) = ξ2
0, where ξ0 is the magnitude of the initial velocity, so that
we have
weβs
= −2gI + ξ2
0 , (3.34)
where I is given by
I =
s
0
sin
αˆs2
2
eβˆs
dˆs ,
which must be determined. We note that the introduction of the limits has required the
use of a dummy variable, ˆs.
To calculate I, we first note that
sin
αˆs2
2
= Im e
i αˆs2
2
= Re −ie
i αˆs2
2
.
Therefore, I can be written as
I = Re −i
s
0
e
i αˆs2
2
+βˆs
dˆs .
Next, we complete the square for the power of the exponential in the integrand, giving
I = Re −ieiβ2
2α
s
0
e
i α
2 (ˆs+ β
iα )
2
dˆs .
Anticipating the appearance of the Fresnel integrals, we write
√
π√
2
t = ˆs + β
iα
√
α√
2
,
so that dˆs =
√
π√
α
dt. Substituting this into I, gives
I =
√
π
√
α
Re −iei β2
2α
√
α
√
π
s
0
ei(π
2
t2
)dt .
It follows that
I =
√
π
√
α
Re −iei β2
2α
√
α
√
π
s
0
cos
π
2
t2
dt + i
√
α
√
π
s
0
sin
π
2
t2
dt ,
28
where the two integrals appearing in I are the Fresnel integrals defined in Section 3.1,
given by (3.11) and (3.12) respectively. Thus, by recalling the shorthand notation for
these integrals defined in Section 3.1, we may write
I =
√
π
√
α
Re −i cos
β2
2α
+ i sin
β2
2α
C
√
α
√
π
s + iS
√
α
√
π
s .
Taking the real part of this and substituting into equation (3.34), we obtain an equation
for w, which is given by
w = −2ge−βs
√
π
√
α
cos
β2
2α
S
√
α
√
π
s + sin
β2
2α
C
√
α
√
π
s + ξ2
0 . (3.35)
We can now apply the final condition to this equation, to obtain an equation which
determines the critical value of ξ0. This final condition is given by ξ(s∗) = 0. Hence,
in terms of the new variable, w, the final condition becomes w(s∗) = 0. Applying this
condition to equation (3.35), we find that the equation for the critical magnitude of the
initial velocity is given by
ξ2
0 = 2g
√
π
√
α
e
−β
√
2π√
α cos
β2
2α
S(
√
2) + sin
β2
2α
C(
√
2) . (3.36)
By setting µ = 0, so that β = 0, we arrive back at equation (3.25), which is the initial
velocity with air resistance neglected, adding confidence that this result is correct.
4 Circular Helix Loop
So far, all calculations have been restricted to two dimensions. Although useful, these
calculations do not resemble the complete characteristics of a vertical loop. In two
dimensions, a vertical loop self intersects, whereas in three dimensions, the track does
not intersect. Therefore, to obtain more realistic values for the initial velocity and g-
force, we must consider the circular loop and the clothoid loop in three dimensions. We
will begin by considering the three-dimensional circular loop, also known as the circular
helix loop.
4.1 Defining the Loop
We can describe a two-dimensional circular loop in the x-z plane using two parametric
equations, which are functions of a variable, t. For a loop of radius a which is centered
at (0, a), these two equations are given by x(t) = a sin(t) and z(t) = −a cos(t) + a. In
addition to these equations, we can introduce a third parametric equation, which adds
29
a linear variance in the y-direction, so that the loop becomes three-dimensional. This
linear variance can be written as y(t) = bt, where b represents the pitch of the helix.
Restricting 0 ≤ t ≤ 2π, the circular helix loop is therefore described parametrically by
x(t) = a sin t , y(t) = bt , z(t) = −a cos t + a . (4.1)
For a real life roller coaster, the distance between the entrance and exit of a vertical
loop is usually between one and two meters. This distance is kept small so that the
loop doesn’t lose its vertical aspect. For this reason, we can therefore assume that old
fashioned circular vertical loops also had this property.
Given that the distance between the entrance and exit of the circular helix loop, d,
is given by d = |y(2π) − y(0)|, we have that b must be in the interval 1
2π ≤ b ≤ 1
π , to
ensure that the entrance and exit of the vertical loop are between one and two meters
apart. The value of b that is chosen from this interval is largely dependent on the overall
size of the loop. We recall the vertical loop element of the Full Throttle roller coaster,
introduced in Section 2.1, which has a radius of a = 19.375m. To model this loop as a
circular helix loop, we will use b = 1
π , giving a distance of 2m between the entrance and
exit of the loop.
4.2 Frenet-Serret Formulas
To help us understand motion in three dimensions, we will derive and make use of the
Frenet-Serret formulas. Consider a general curve, C, in three-dimensional space with
arc length, s (see Figure 10). We define r(s) as the position vector of a point on the
curve, ˆt as the tangential unit vector pointing in the direction of motion and ˆn as the
unit normal vector. Since ˆt and ˆn are orthogonal unit vectors, they satisfy ˆt · ˆn = 0.
Then, we introduce a new unit vector known as the binormal unit vector, ˆb, which is
defined as ˆb = ˆt × ˆn [5].
The Frenet-Serret formulas are a set of relations between the three unit vectors,
defining how each unit vector varies with arc length. Defining κ as the curvature, we
know from (3.7) and (3.8) that
dˆt
ds
= κˆn . (4.2)
Next we differentiate ˆb with respect to s, so that we obtain
dˆb
ds
=
d
ds
(ˆt × ˆn) =
dˆt
ds
× ˆn + ˆt ×
dˆn
ds
. (4.3)
Noting that ˆn × ˆn = 0, we substitute (4.2) into (4.3) to give
dˆb
ds
= ˆt ×
dˆn
ds
.
30
Figure 10: General curve, C, in three dimensional space, with the position vector for a
point on C given by r(s) and the tangential, normal and binormal unit vectors given by
ˆt, ˆn and ˆb respectively.
From the definition of a vector product, we know from this equation that dˆb
ds is orthogonal
to ˆt. We also know that ˆb · ˆb = 1. Therefore,
dˆb
ds
· ˆb = 0 =
dˆb
ds
· ˆt .
This implies that dˆb
ds must be in the direction of ˆn. Thus, defining the magnitude of dˆb
ds
as −τ, we have
dˆb
ds
= −τ ˆn . (4.4)
The variable τ is known as the torsion of a curve. Torsion is a measure of the rate at
which a curve is twisting out of its osculating plane at a point [3].
Next, we differentiate the unit vector ˆn, with respect to s. Since ˆb = ˆt× ˆn, we know
from use of the right hand rule that ˆn = ˆb × ˆt and ˆt = ˆn × ˆb. Hence, by differentiating
ˆn with respect to s and using (4.2) and (4.4), we obtain
dˆn
ds
=
d
ds
(ˆb × ˆt) =
dˆb
ds
× ˆt + ˆb ×
dˆt
ds
= −τ ˆn × ˆt + ˆb × κˆn = −κˆt + τˆb . (4.5)
The combination of equations (4.2), (4.4) and (4.5) is known as the Frenet-Serret Triad:



dˆt
ds = κˆn
dˆb
ds = −τ ˆn
dˆn
ds = −κˆt + τˆb
(4.6)
31
These equations can be found in [5] and [3]. By defining a position vector for a point
on a three-dimensional curve, we can use the Frenet-Serret Triad and other geometry
definitions to calculate ˆt, ˆn and ˆb, along with κ and τ, which can then be used to derive
the governing equations of motion along the curve.
4.3 Motion Around the Circular Helix Loop
Using the definition of a circular helix loop, we now define a position vector for a point on
the curve. Using the Frenet-Serret Triad and other definitions, we can use this position
vector to calculate ˆt, ˆn and ˆb, along with κ and τ, which will allow us to study the
motion of a point mass around the loop.
We begin by defining r(t) as the position vector of a point along the curve. Using
(4.1), we can write
r(t) = a sin tˆi + btˆj + (−a cos t + a)ˆk . (4.7)
Next we parameterize this position vector, so that it is given in terms of arc length, s.
Arc length is given by [3]
s =
dr
dt
dt =
dx
dt
2
+
dy
dt
2
+
dz
dt
2
dt . (4.8)
Therefore, substituting (4.7) into (4.8) gives
s = (a cos t)2 + b2 + (a sin t)2dt = a2 + b2dt = t a2 + b2 .
Hence, by defining λ =
√
a2 + b2, (4.7) can be written as
r(s) = a sin
s
λ
ˆi +
bs
λ
ˆj + (−a cos
s
λ
+ a)ˆk .
Now that the position vector is in terms of arc length, we can use equation (3.1) to
calculate ˆt. Differentiating r with respect to s, we find that
ˆt =
a
λ
cos
s
λ
ˆi +
b
λ
ˆj +
a
λ
sin
s
λ
ˆk . (4.9)
We now use the first equation of the Frenet-Serret Triad, also given by (4.2), to calculate
ˆn and κ. Differentiating ˆt with respect to s, we obtain
κ =
a
λ2
, ˆn = − sin
s
λ
ˆi + cos
s
λ
ˆk . (4.10)
Now that we have defined ˆt and ˆn, we can apply them to the definition of the binormal
unit vector, so that we can obtain ˆb. Calculating ˆt × ˆn, we have
ˆb =
b
λ
cos
s
λ
ˆi −
a
λ
ˆj +
b
λ
sin
s
λ
ˆk . (4.11)
32
Finally, we use the second equation of the Frenet-Serret Triad, also given by (4.4), along
with equation (4.10), to calculate the torsion. By differentiating ˆb with respect to s, we
find that the torsion is given by
τ = −
b
λ2
.
This completes the calculation of each component of the Frenet-Serret Triad. A useful
check to add confidence that these calculations are correct is to substitute into the final
equation of the Triad, also given by (4.5), and check that this equation holds.
Ignoring air resistance and friction, we now apply Newton’s second law to the mass
and resolve in the direction of the three unit vectors. This will give us a governing equa-
tion of motion, which can be solved to find an equation which calculates the magnitude
of the initial velocity that is required for the mass to successfully reach the top of a
circular helix loop. Given that air resistance and friction are neglected and the loop is
circular, by use of the energy argument in Section 3.4, we expect the equation for this
magnitude to be identical to the two-dimensional equation, given by (2.13).
We begin by summing the forces acting on the mass and use equation (3.15), so that
Newton’s second law gives
R − mgˆk = m ˆt
d2s
d2t
+ ˆnκ
ds
dt
2
. (4.12)
Now that we are working in three-dimensional space, we must take into consideration
that the reaction force, R, will also have a component in the ˆb-direction. We write
R = Rn ˆn + Rb
ˆb , (4.13)
where Rn and Rb represent the magnitude of the reaction force in the ˆn- and ˆb-direction
respectively. Hence, by substituting (4.13) into equation (4.12) and taking the dot
product of this equation with ˆt, ˆn and ˆb, we obtain
ˆt · (4.12) : −mg
a
λ
sin
s
λ
= m
d2s
dt2
, (4.14)
ˆn · (4.12) : Rn − mg cos
s
λ
= mκ
ds
dt
2
, (4.15)
ˆb · (4.12) : Rb − mg
b
λ
sin
s
λ
= 0 . (4.16)
Equations (4.15) and (4.16) can be rearranged to give an equation to calculate the
magnitude of the reaction force in the ˆn- and ˆb-direction respectively. Equation (4.14)
is a second order, ordinary differential equation. This can be integrated so that we can
33
find an equation for the magnitude of the mass’ velocity at any point on the circular
helix loop.
To integrate equation (4.14), we introduce the variable ξ = ds
dt , so that d2s
dt2 = ξ dξ
ds .
Substituting this into (4.14), gives a separable equation. By separating the variables,
integrating by substitution and determining the constant of integration by use of the
initial condition, ξ(0) = ξ0, where ξ0 is the magnitude of the initial velocity, we obtain
1
2
ξ2
= ag cos
s
λ
+
1
2
ξ2
0 − ag . (4.17)
Now that we are working in three dimensions, we require a new final condition. At
the top of the circular helix loop, s = λπ. Therefore, the new final condition is given by
ξ(λπ) = 0. Applying this condition to equation (4.17), we find that
ξ0 = 2
√
ag , (4.18)
which is exactly equation (2.13), as expected.
4.4 G-Force
In Section 2.4, we proved that the magnitude of g-force experienced was too large, as a
point mass travelled around a two-dimensional circular loop. Although the circular helix
loop is also circular in shape, it is useful to see how adding variance in the y-direction to
the problem affects the g-force experienced, compared to the two-dimensional problem
in Section 2.4.
Ignoring air resistance and friction, we substitute equations from Section 4.3 into
equation (2.26). This gives us an equation for the magnitude of g-force, G, for which we
will calculate the turning points and classify whether these are minimum or maximum
turning points. The maximum magnitude of g-force, Gmax, can then be calculated, so
that we can determine whether the circular helix is an acceptable shape for a vertical
loop.
Recalling the equation for the magnitude of g-force given by (2.26), we substitute
(4.14) and (4.17) into (2.26) and recall from (4.10) that κ = a
λ2 for the circular helix
loop, so that
G =
1
g
−
ag
λ
sin
s
λ
ˆt +
a
λ2
2ag cos
s
λ
− 1 + ξ2
0 ˆn .
This is the same as writing
G =
1
g
a2g2
λ2
sin2 s
λ
+
a2
λ4
2ag cos
s
λ
− 1 + ξ2
0
2
1
2
. (4.19)
34
Next we calculate the turning points of this equation. Following the method from
Section 2.4, we square equation (4.19) and differentiate with respect to s to give
2G
dG
ds
= sin
s
λ
cos
s
λ
2a2
λ3
−
8a4
λ5
+ sin
s
λ
8a4
λ5
−
4a3ξ2
0
gλ5
. (4.20)
The turning points are given by dG
ds = 0. This occurs when s = nλπ, where n ∈ N, or
when
cos
s
λ
=
8a4
λ5
−
4a3ξ2
0
gλ5
= 0 . (4.21)
Using equation (4.18), we make the substitution ξ0 = 2
√
ag and find that equation (4.21)
cannot be satisfied. Therefore, the only turning points are given by s = nλπ. Given
that 0 ≤ s ≤ λπ for the first half of the circular helix loop, we neglect turning points
that are not within this interval. Hence, the turning points of equation (4.19) that we
are interested in are s = 0 and s = λπ.
By differentiating equation (4.20), we can classify these turning points. We define
two constants:
β =
2a2
λ3
−
8a4
λ5
, γ =
8a4
λ5
−
4a3ξ2
0
gλ5
,
and then differentiate (4.20) with respect to s, to give
2
dG
ds
2
+ 2G
d2G
ds2
=
β
λ
cos2 s
λ
−
β
λ
sin2 s
λ
+
γ
λ
cos
s
λ
.
By substituting the two turning points into this equation, we find that
s = 0, dG
ds = 0 2Gd2G
ds2 = β
λ + γ
λ < 0 → maximum
s = λπ, dG
ds = 0 2Gd2G
ds2 = −β
λ − γ
λ > 0 → minimum
Since s = 0 is the only maximum turning point, substituting s = 0 into equation (4.19)
will give the maximum magnitude of g-force that is experienced. Recalling the Full
Throttle example in Section 2.1, we take a = 19.375m, ξ0 = 70mph and recall from
Section 4.3 that λ =
√
a2 + b2, where b = 1
π . Substituting these values and s = 0 into
equation (4.19), we find that Gmax ≈ 5.15g. Because Gmax > 5g, the circular helix loop
is not an acceptable shape for a vertical loop.
By comparing this maximum value of Gmax with the value calculated for the two-
dimensional circular loop in Section 2.4, we see that the three-dimensional circular loop
produces approximately the same value of Gmax. However, since G is dependent upon
the pitch, b, this only holds for certain values of b.
When b = 0, the vertical loop becomes two-dimensional and the maximum value
of g-force will be equal to the value calculated in Section 2.4. For b = 0, the vertical
35
loop becomes three-dimensional and we find that as b increases, the g-force experienced
decreases. Hence, by increasing b, it would be possible to reduce the g-force so that
Gmax < 5g. However, as previously discussed in Section 4.1, when b becomes larger
than a certain value (typically b > 1
π for a circular helix loop), the loop loses its vertical
aspect and therefore b cannot be increased. Because of this restriction on b, we can
conclude that Gmax always has approximately the same value for the two-dimensional
and three-dimensional circular loop. By applying properties of the clothoid curve, we
can produce a three-dimensional clothoid helix loop, in an attempt to reduce this value
to a tolerable amount.
5 Clothoid Helix Loop
In two dimensions, the first half of a clothoid loop is defined parametrically by the
Fresnel integrals for 0 ≤ s ≤ s∗, with the second half being generated by reflection, as
seen in Section 3.1. After redefining these equations, we can introduce a third parametric
equation, which adds a linear variance in the y-direction, so that the loop becomes three-
dimensional. This three-dimensional loop is known as the clothoid helix loop.
We begin by redefining the two Fresnel integrals in terms of the variables t and ζ,
such that
x(t) =
√
π
√
α
√
α
√
π
t
0
cos
π
2
ζ2
dζ , z(t) =
√
π
√
α
√
α
√
π
t
0
sin
π
2
ζ2
dζ .
The linear variance in the y-direction can be written as y(t) = bt, where b is the pitch
of the helix. The combination of these three parametric equations, in addition with
the condition 0 ≤ t ≤ t∗, defines the first half of the clothoid helix loop. In shorthand
notation, these equations are
x(t) =
√
π
√
α
C
√
α
√
π
t , y(t) = bt , z(t) =
√
π
√
α
S
√
α
√
π
t . (5.1)
For the circular helix in Section 4.1, we set b = 1
π , giving a distance of 2m between the
entrance and exit of the loop. So that we are consistent with the distance between the
entrance and exit of the two loop shapes, we will choose b so that the clothoid helix
loop also has a 2m separation between its entrance and exit. Since the second half of a
clothoid helix loop is generated by reflection, t∗ = 2
√
2π√
α
for the full clothoid helix loop.
Therefore, we set b =
√
α√
2π
.
36
5.1 Motion Around the Clothoid Helix Loop
We now consider the motion of a point mass around the clothoid helix loop defined
by (5.1). We look to calculate an equation to determine the magnitude of the initial
velocity that is required for the mass to just reach the top of the vertical loop. We
will achieve this by calculating all components of the Frenet-Serret Triad and applying
Newton’s second law to the mass, with air resistance and friction neglected, so that
an equation of motion can be found. When solving this equation, we expect to find
a critical magnitude equation that is identical to the equation for the two-dimensional
clothoid loop given by (3.25), because of the energy argument in Section 3.4.
We begin by using (5.1) to define the position vector of a point on the loop, r(t),
such that
r(t) =
√
π
√
α
C
√
α
√
π
t ˆi + btˆj +
√
π
√
α
S
√
α
√
π
t ˆk . (5.2)
Next, we parameterize (5.2), so that r is in terms of arc length, s. Using (4.8), we have
s = cos2 αt2
2
+ b2
+ sin2 αt2
2
1
2
dt = (1 + b2
)
1
2 dt ,
giving t = s
λ , where λ = (1 + b2). Therefore, (5.2) becomes
r(t) =
√
π
√
α
C
√
α
√
π
s
λ
ˆi + btˆj +
√
π
√
α
S
√
α
√
π
s
λ
ˆk . (5.3)
Now that the position vector is in terms of s, we can use the Frenet-Serret Triad
and other geometry definitions to calculate the unit vectors ˆt, ˆn, and ˆb. Along with
these unit vectors, we can also calculate the torsion and curvature, given by τ and κ
respectively.
We start by using equation (3.1) to calculate ˆt. Differentiating (5.3) with respect to
s, we find that
dˆr
ds
= ˆt =
1
λ
cos
αs2
2λ2
ˆi +
b
λ
ˆj +
1
λ
sin
αs2
2λ2
ˆk . (5.4)
Using this equation for ˆt and the first equation from the Frenet-Serret Triad, we can
then calculate ˆn and κ. Differentiating (5.4) with respect to s, we obtain
κ =
αs
λ3
, ˆn = − sin
αs2
2λ2
ˆi + cos
αs2
2λ2
ˆk . (5.5)
Next we use the definition of the binormal unit vector to find ˆb. Using (5.4) and (5.5)
to calculate ˆt × ˆn, we find that ˆb is given by
ˆb =
b
λ
cos
αs2
2λ2
ˆi −
1
λ
ˆj +
b
λ
sin
αs2
2λ2
ˆk . (5.6)
37
Finally, we use the second equation of the Frenet-Serret Triad and equation (5.5) to
calculate the torsion. By differentiating (5.6) with respect to s, we calculate that
τ = −
bαs
λ3
.
This completes the calculation of all components of the Frenet-Serret Triad.
We now apply Newton’s second law to the mass and resolve in the direction of the
three unit vectors. Ignoring friction and air resistance and recalling the equation for the
reaction force, R, in three dimensions given by (4.13), Newton’s second law gives
Rn ˆn + Rb
ˆb − mgˆk = m ˆt
d2s
d2t
+ ˆnκ
ds
dt
2
. (5.7)
Taking the dot product of this equation with ˆt, ˆn and ˆb, we have
ˆt · (5.7) : −
mg
λ
sin
αs2
2λ2
= m
d2s
dt2
, (5.8)
ˆn · (5.7) : Rn − mg cos
αs2
2λ2
= κ
ds
dt
2
, (5.9)
ˆb · (5.7) : Rb − mg cos
αs2
2λ2
= 0 . (5.10)
By introducing the variable ξ = ds
dt , so that via the chain rule, d2s
dt2 = ξ dξ
ds , we substi-
tute ξ into equation (5.8) and attempt to find an equation to determine the magnitude
of the initial velocity that is required for a mass to just reach the top of the loop.
Substituting ξ into (5.8), we have
ξ
dξ
ds
= −
g
λ
sin
αs2
2λ2
.
Introducing the dummy variable ˆs, we separate the variables and integrate this equation
between zero and s in anticipation of the initial condition, ξ(0) = ξ0, where ξ0 is the
magnitude of the initial velocity. This gives
1
2
ξ2
= −
g
λ
s
0
sin
αˆs2
2λ2
dˆs +
1
2
ξ2
0 . (5.11)
Then, in anticipation of the appearance of a Fresnel integral, we write ˆs = λ
√
π√
α
t, so that
dˆs = λ
√
π√
α
dt. Therefore, equation (5.11) becomes
1
2
ξ2
= −g
√
π
√
α
1
λ
√
α
√
π
s
0
sin
π
2
t2
dt +
1
2
ξ2
0 , (5.12)
38
To obtain an equation for the critical value of ξ0, a new final condition is now
required. The arc length at the top of the vertical loop, s∗, is given by the first value of
s, excluding s = 0, where dz
ds = 0. Therefore, given that
z =
s
0
sin
αˆs2
2λ2
dˆs ,
we have
dz
dˆs
= sin
αˆs2
2λ2
= 0 ,
giving s∗ = λ
√
2π√
α
. Thus, the final condition is given by ξ = 0, at s = λ
√
2π√
α
. Applying
this final condition to equation (5.12), we find that the equation determining the critical
value of ξ0 is given by
ξ2
0 = 2g
√
π
√
α
S(
√
2) ,
which is exactly (3.25), as expected.
5.2 G-Force
In Section 3.5, we discovered that using a clothoid loop instead of a circular loop reduces
the magnitude of g-force experienced when travelling around the loop, to a tolerable
amount. Using equations from Section 5.1 where air resistance and friction were ne-
glected, we can verify that the three-dimensional clothoid helix loop also reduces the
magnitude of g-force, along with comparing how the g-force varies compared to the
two-dimensional clothoid loop.
The magnitude of g-force, G, is given by equation (3.27), such that
G =
1
g
d2s
dt2
ˆt + κ
ds
dt
2
ˆn .
Given that ξ = ds
dt , we substitute equations (5.8) and (5.12) into G, whilst noting that
for a clothoid helix loop, κ = αs
λ3 , where λ =
√
1 + b2, so that we obtain
G =
1
g
g
λ
sin
αs2
2λ2
ˆt +
αs
λ3
−2g
√
π
√
α
S
1
λ
√
α
√
π
s + ξ2
0 ˆn .
This is equivalent to writing
G =
1
g
g2
λ2
sin2 αs2
2λ2
+
α2s2
λ6
−2g
√
π
√
α
S
1
λ
√
α
√
π
s + ξ2
0
2
1
2
. (5.13)
By differentiating equation (5.13) with respect to s and setting dG
ds = 0, we can
calculate the turning points. These can then be classified by taking the second derivative
of G with respect to s and substituting in the turning points.
39
Due to the complexity of equation (5.13), we differentiate using Maple. So that
we can compare the three-dimensional clothoid loop with the two-dimensional clothoid
loop, we will use the same size vertical loop from the roller coaster Full Throttle for
the three-dimensional clothoid loop, as we did for the two-dimensional clothoid loop in
Section 3.3. Noting that α is not affected by the problem becoming three-dimensional,
we take α ≈ 1.07 × 10−3 and ξ0 = 70mph, with b =
√
α√
2π
. Differentiating equation (5.13)
once with respect to s and setting dG
ds = 0, we find that the turning points of (5.13) are
given by s = 0, s = s∗ and s ≈ 40.76m. Differentiating again, we classify s = 0 and
s = s∗ as minimum turning points and s ≈ 40.76m as a maximum turning point. Given
that we only have one maximum, we substitute the non-truncated value of s ≈ 40.76m
into equation (5.13), giving Gmax ≈ 2.50g, which is sufficiently small enough to make
the vertical loop safe.
Comparing the truncated value of s where the g-force is maximum with the truncated
value of s found in Section 3.5 for the two-dimensional clothoid loop, it can be seen that
the maximum g-force occurs at approximately the same point on the three-dimensional
loop as it does on the two-dimensional loop. Similarly, the truncated values of Gmax
are approximately the same in two and three dimensions. This shows that increasing
the dimensions of the problem has minimal effect on the g-force and the point where it
takes its maximum value. As previously discussed in Section 4.4, since equation (5.13)
is dependent upon b, it is possible to reduce Gmax further by increasing b. However,
this leads to the vertical loop losing its vertical aspect and because Gmax is already
sufficiently small enough, increasing b is not required.
6 Multi-Mass Problem
Although some roller coaster trains are made up of one carriage, the majority of roller
coasters use a train made up of five to seven carriages. As an extension to the problems
we have seen so far, we can consider a multi-mass system, where multiple point masses
are joined together, forming a train.
Beginning with two connected masses on a two-dimensional circular loop, an equation
of motion can be derived for the two masses. Then, by adding further masses to the
train, a formula calculating an equation of motion for n masses, where n ∈ N, can be
determined. By analysing this equation, the effects of adding extra masses to the train
can be discovered. For simplicity, air resistance and friction will be neglected.
40
6.1 Two-Mass Train
Consider two point masses, m1 and m2, where m1 = m2, that are connected by a light,
inextensible rod, on a two-dimensional circular loop of radius a (see Figure 11). The
tension in the rod is given by T , with magnitude T. We define θ as the angle between
the negative z-axis and m2, with β being the constant angular separation of m1 and
m2. The transverse unit vector is defined as ˆθi and the radial unit vector is defined as
ˆri, where i = 1, 2, defining which mass the unit vectors correspond to. Similarly, the
position vector of each mass is given by ri and the reaction force for each mass is given
by Ri, with magnitude Ri. It should be noted that 0 < β < π. This condition implies
that when θ = 0, the first mass in the train has not reached the top of the loop.
Figure 11: Problem setup for a two-mass train on a circular loop.
We apply Newton’s second law to each mass individually, starting with m1. The total
angle at any point between the negative z-axis and m1 is given by θ + β. Recalling the
equation for the acceleration of a mass in polar coordinates given by (2.4) and defining
ψ = θ + β, we apply Newton’s second law to m1, which gives
−ma
dψ
dt
2
ˆr1 + ma
d2ψ
dt2
ˆθ1 = −mg sin ψˆθ1 − R1ˆr1 − T ˆx , (6.1)
where ˆx is a unit vector pointing in the direction of the tension. Given that r1 = aˆr1
and r2 = aˆr2, we can write ˆx as
ˆx =
r2 − r1
|r2 − r1|
=
a(ˆr2 − ˆr1)
|ˆr2 − ˆr1|
.
By calculating |ˆr2 − ˆr1|2 and using the definition of the dot product, it can be shown
41
that
ˆx =
a(ˆr2 − ˆr1)
2 sin β
2
.
Next, we apply Newton’s second law to m2, noting that the angle between this mass
and the negative z-axis is θ. This gives
−ma
dθ
dt
2
ˆr2 + ma
d2θ
dt2
ˆθ2 = −mg sin θˆθ2 − R2ˆr2 + T ˆx . (6.2)
We wish to solve in the direction of motion for each mass. Therefore, we calculate
(6.1) · ˆθ1 and (6.2) · ˆθ2. So that we can calculate ˆx · ˆθi, we convert ˆθi and ˆri into
Cartesian coordinates, such that
ˆri = sin(φ)ˆi + cos(φ)ˆj , (6.3)
ˆθi = − cos(φ)ˆi + sin(φ)ˆj , (6.4)
where φ is the total angle between the negative z-axis and the mass in question. Calcu-
lating the dot products, we find that
(6.1) · ˆθ1 : ma
d2ψ
dt2
= −mg sin ψ −
Ta sin β
2 sin β
2
, (6.5)
(6.2) · ˆθ2 : ma
d2θ
dt2
= −mg sin θ +
Ta sin β
2 sin β
2
. (6.6)
Since β is constant, d2ψ
dt2 = d2θ
dt2 . Therefore, by adding equations (6.5) and (6.6) and using
a trigonometric identity, we have
d2θ
dt2
+
ˆg2
a
sin θ +
β
2
= 0 , (6.7)
where ˆg2 = g cos β
2 .
Clearly if β = 0, we obtain the equation of motion for a single mass train given by
(2.9), adding confidence that this equation is correct. Equation (6.7) also tells us that
adding an extra mass to the train reduces the effect of gravity that it experiences, where
ˆg2 is this new gravitational constant, which is dependent on β. In addition, we note that
the angle β
2 is the midpoint of the arc between m1 and m2, indicating that the train
behaves as though one mass is positioned at the midpoint between the two masses on
the circle.
42
6.2 Three-Mass Train
We now add a third mass, m3, to the train and attempt to find an equation of motion
in a similar way to the two-mass problem in the previous section. We attach m3 to m2
by a light, inextensible rod, so that θ is now the angle between the negative z-axis and
m3 and β is the constant angular separation between each mass, where 0 < 2β < π
(see Figure 12). The radial unit vector of each mass is denoted ˆri and the transverse
unit vector is denoted ˆθi, where i = 1, 2, 3. Similarly, the position vector of each mass is
given by ri and the reaction force is given by Ri, with magnitude Ri. The magnitudes of
the tension in the rod connecting m1 and m2 are denoted T12 and T21, where T12 = T21.
Likewise, the magnitudes of the tension in the rod connecting m2 and m3 are denoted
T23 and T32, where T23 = T32.
Figure 12: Problem setup for a three-mass train on a circular loop.
We apply the same approach as seen in Section 6.1, starting with m1. Given that
the total angle at any point between the negative z-axis and m1 is given by θ + 2β, we
define η = θ + 2β, so that Newton’s second law gives
−ma
dη
dt
2
ˆr1 + ma
d2η
dt2
ˆθ1 = −mg sin ηˆθ1 − R1ˆr1 − T12 ˆx , (6.8)
where ˆx is a unit vector pointing in the direction of T21. Similarly, by defining ψ = θ+β
and applying Newton’s second law to m2, we obtain
−ma
dψ
dt
2
ˆr2 + ma
d2ψ
dt2
ˆθ2 = −mg sin ψˆθ2 − R2ˆr2 + T21 ˆx − T23 ˆy , (6.9)
43
where ˆy is a unit vector pointing in the direction of T32. Finally, applying Newton’s
second law to m3, we have
−ma
dθ
dt
2
ˆr3 + ma
d2θ
dt2
ˆθ3 = −mg sin θˆθ3 − R3ˆr3 + T32 ˆy . (6.10)
Following the method discussed in Section 6.1, it can be shown that ˆx and ˆy are
given by
ˆx =
a(ˆr2 − ˆr1)
2 sin β
2
, ˆy =
a(ˆr3 − ˆr2)
2 sin β
2
.
Substituting ˆx and ˆy into (6.8), (6.9) and (6.10) and recalling the definitions of ˆri and
ˆθi in Cartesian coordinates, which are given by (6.3) and (6.4) respectively, we take the
dot product of equations (6.8), (6.9) and (6.10) with their corresponding transverse unit
vector. This gives
(6.8) · ˆθ1 : ma
d2η
dt2
= −mg sin η −
T12a sin β
2 sin β
2
, (6.11)
(6.9) · ˆθ2 : ma
d2ψ
dt2
= −mg sin ψ +
T21a sin β
2 sin β
2
−
T23a sin β
2 sin β
2
, (6.12)
(6.10) · ˆθ3 : ma
d2θ
dt2
= −mg sin θ +
T32a sin β
2 sin β
2
. (6.13)
Since β is a constant, d2η
dt2 = d2ψ
dt2 = d2θ
dt2 . Therefore, by adding equations (6.11), (6.12)
and (6.13) and applying a trigonometric identity, we obtain
d2θ
dt2
+
ˆg3
a
(sin(θ + β)) = 0 , (6.14)
where ˆg3 = g
3 (1 + 2 cos 2β).
Again, if β = 0, we obtain the equation of motion for a single mass train given
by (2.9). The three-mass train also experiences reduced gravity, where ˆg3 is the new
gravitational constant, which is also dependent on β. Since the angular separation of
the masses must be within the interval 0 < 2β < π, we have ˆg3 > ˆg2. Thus, the addition
of a third mass has further reduced the effects of gravity. To generalise the behaviour
of g as the number of masses in the train increases, we can calculate a formula for ˆgn,
where n ∈ N is the number of masses in the train.
6.3 N-Mass Train
Applying ideas and methods from Sections 6.1 and 6.2, we can easily derive equations of
motion for a four, five and six-mass train, which can be used in conjunction with (2.9),
44
(6.7) and (6.14), to derive a general formula calculating the equation of motion for an
n-mass train, where n ∈ N. This formula is given by
d2θ
dt2
+
ˆgn
a
sin θ +
(n − 1)
2
β = 0 . (6.15)
Here ˆgn represents the reduced gravitational constant, θ is the angle between the negative
z-axis and mn and β is the constant angular separation of the masses, where 0 <
(n − 1)β < π. We find that ˆgn is dependent upon whether n is odd or even, such that
n is odd : ˆgn =
g
n
(1 + (n − 1) cos 2β) , (6.16)
n is even : ˆgn =
2g
n
1
2
n−1
i=0
cos
β
2
+ iβ . (6.17)
For a roller coaster, a typical value of n is given by 1 ≤ n ≤ 7. Using (6.16) and
(6.17), we can calculate ˆg1 up to ˆg7 and plot how these vary with β. This allows us to
analyse how adding extra masses to the train varies the effects of gravity (see Figure 13
and Figure 14).
Figure 13: Graph showing how ˆgn varies with β, for 0 < β < π
6 and 1 ≤ n ≤ 7, where
n ∈ N.
From Figure 13, we can see that ˆg3 > ˆg4 for 0 < β < π
6 , meaning the addition of
an extra mass to the three mass train has not further reduced the gravity experienced.
Therefore, given that we found that ˆg3 > ˆg2, it is not possible to draw up any conclusion
about the behaviour of ˆgn, for general n.
45
Figure 14: Left: graph showing how ˆgn varies with β, where n = 2k, 1 ≤ 2k ≤ 7, and
k ∈ N, for 0 < β < π
6 . Right: graph showing how ˆgn varies with β, where n = 2k − 1,
1 ≤ 2k − 1 ≤ 7, and k ∈ N, for 0 < β < π
6 .
However, if we consider separate cases for n odd and n even, we can see from Figure
14 that increasing n, decreases the value of ˆgn. From Figure 13 and Figure 14, it is
also clear that decreasing the value of β, increases the value of ˆgn. Therefore, given the
restriction 0 < (n − 1)β < π, if extra masses are continually added to the train, the
angular separation between them must subsequently decrease, so that the effect that
each action has on the value of ˆgn should eventually cancel out.
We can prove this result by letting n become large and subsequently decreasing β.
Let β(n − 1) = φ (φ constant), so that as n → ∞, β → 0. When n is odd, clearly as
β → 0, (1 + (n − 1) cos 2β) → n, so that ˆgn → g. For the case when n is even, we let
n = 2k, where k ∈ N, so that
lim
β→0
k−1
i=0
cos
β
2
+ iβ = k =
n
2
.
Hence, ˆgn → g. Therefore, by adding more masses to the train and subsequently reduc-
ing the angle between them, the reduction in g continually decreases, until there is no
reduction whatsoever.
Using this result, we note that as n → ∞ and β → 0, equation (6.15) becomes
d2θ
dt2
+
g
a
sin θ +
φ
2
= 0 ,
where the angle φ
2 is the midpoint of the arc between the first and last mass of the train.
This implies that the train will behave as though a single mass is positioned at this
46
midpoint. It should be noted that this midpoint signifies the center of mass for all of
the combined point masses.
Using the formulas for ˆgn, we can compute the reduced gravitational constant for
a real life roller coaster. We recall the roller coaster Full Throttle that was introduced
in Section 2.1. This roller coaster consists of three cars per train. Assuming that the
vertical loop is circular, we use equation (6.16) to calculate ˆg3. Taking into consideration
the size of the loop and that β would be small for a real life roller coaster, we approximate
the angular separation of the cars as β = π
16, so that ˆg3 ≈ 9.31ms−2. Since g = 9.81ms−2,
it is clear from this calculation that for a realistic setup, the reduction in the effects of
gravity are minimal.
7 Conclusion
From the calculations we have seen, it is evident that the mathematics behind the design
of a roller coaster is incredibly complex. After using trial and error in an attempt to
design new and exciting track layouts, multiple problems arose. Eventually, designers
applied rigorous mathematical techniques to the design process, allowing them to un-
derstand what was causing these particular problems with their designs, before seeking
solutions to the problems. The vertical loop is an excellent example of this process.
The main aim of this project was to replicate the calculations that designers would
have made when designing roller coaster vertical loops, for two different loop shapes; the
circular loop and the clothoid loop. These calculations included finding the magnitude
of the initial velocity that is required for a mass to reach the top of a vertical loop, along
with the g-force experienced as the mass travelled around the loop, so that we could
identify any problems/benefits of using either loop shape.
We began by considering the two-dimensional circular loop (Section 2). The first
calculation that was performed was the derivation of an equation linking the height of
the vertical loop, to the magnitude of the initial velocity, ξ0, that the train must be given
to successfully reach the top of the loop. To simplify this calculation, we used a point
mass instead of a full length train. This simplification was used throughout Sections
2-5. We also began by neglecting drag and friction.
Using polar coordinates, we found an equation of motion for the point mass on the
loop. This equation was then solved and after applying an initial and final condition, we
obtained an equation for the critical value of ξ0. This equation was also found using an
energy argument. As an extension to this calculation, the drag force was included and
47
a similar method was used to find an alternate equation for ξ0. Given the form of the
solution, further adjustments using Taylor/binomial expansions were made, allowing an
equation for ξ0 to be determined where the drag coefficient was small.
After building many roller coasters with circular vertical loops, riders complained
about the uncomfortable experiences they were having travelling around these loops.
To demonstrate this problem, we mathematically defined the g-force directly from its
definition and used equations from Sections 2.1, to find an equation for the magnitude of
g-force, G, that is experienced as the mass travelled around the loop. Using values from
the roller coaster Full Throttle, it was calculated that this circular shape loop produced
a maximum magnitude of g-force which was too high (Gmax ≈ 5.15g), which occurred
as the mass entered and exited the loop. This showed why the circular loop is not an
acceptable shape for a vertical loop.
Next we considered the two-dimensional clothoid loop, which required us to define
the loop parametrically. Starting with the basic definition of a clothoid curve, differential
geometry was used to derive two integrals known as the Fresnel integrals. Parametrically
plotting these integrals produced the clothoid curve. Properties of this curve were then
used to create half of a clothoid loop.
By using techniques similar to those seen in Section 2.1, an equation determining the
critical value of ξ0, with air resistance and friction neglected, was found. By studying
the Full Throttle roller coaster, it was shown that the critical value of ξ0 was identical
to the value for the circular loop. The reasoning behind this result was then confirmed
by an energy argument in Section 3.4. However, the most important calculation was
proving that this loop shape does indeed reduce the g-force experienced. For the Full
Throttle example, we found that the maximum magnitude of g-force experienced was
Gmax ≈ 2.50g, which is slightly less than half of the value for a circular loop of the same
height. Thus, it was proven that the clothoid loop was an acceptable loop shape to be
used.
Next we studied both loop shapes in three dimensions. To understand motion in
three-dimensional space, a binormal unit vector was introduced and the Frenet-Serret
Triad was derived, which was used to help derive an equation for ξ0. Both loop shapes
were found to produce almost identical values of Gmax to their two-dimensional counter-
parts. The equations for ξ0 were also identical to the two-dimensional examples, which
was verified by the energy argument in Section 3.4.
Finally, as an extension to the single mass problem, the behaviour of a train of point
masses travelling around a two-dimensional circular loop, with air resistance and friction
48
neglected, was investigated. It was discovered that a two-mass train reduced the effects
of gravity on the train, which was dependent upon the angular separation of the masses.
As more masses were added, we found that the reduction of gravity was dependant on
whether the number of cars in the train, n, was even or odd. In both cases, for constant
β and 0 < (n − 1)β < π, the reduction in gravity became bigger as the n increased. A
formula to calculate the equation of motion for an n-mass train was then determined,
which included two formulae for the reduced gravitational constant, ˆgn. Whilst studying
the behaviour of ˆgn as n → ∞, so that β → 0, it was seen that ˆgn → g.
Vertical loops are one of a variety of inversions that are found on a roller coaster. To
further the calculations we have seen, we could mathematically define different inversions
and apply these definitions to the equation of motion along a general curve. Then,
for each inversion, the g-force and critical value of ξ0 could be calculated by following
similar methods that have been seen throughout this project for the vertical loop. These
calculations would also require knowledge of other fundamental concepts of roller coaster
design, such as banking the track, which is discussed in [1], pages 47-50. An interesting
example of an inversion that could be studied is the tilt-incline loop; a vertical loop that
is tilted at an angle (see Figure 15).
Figure 15: Tilt-incline loop; an inversion found on The Swarm, at Thorpe Park [11].
Inversions, particularly vertical loops, are included in track layouts for hundreds of
roller coasters which are operational today. These inversions are used to maximise the
thrill and experience of riding a roller coaster. As designers continue to invent taller
and faster roller coasters, along with new and exciting inversions, mathematics is vitally
important to prove that the passengers will be safe when riding these roller coasters. It is
these new inventions that encourage people to visit theme parks every year, highlighting
the important role that mathematics plays in the industry and the future construction
of all roller coasters.
49
References
[1] Weisenberg, N. (2013) Coasters 101- An Engineer’s Guide to Roller Coaster Design.
3rd Edition. pp 11-54
[2] Abramowitz, M. and Stegun, I.A. (1972) Handbook of Mathematical Functions with
Formulas, Graphs and Mathematical Tables. 10th Edition. New York: Dover Publi-
cations. pp 300-301
[3] Widder, D.V. (1961) Advanced Calculus. 2nd Edition. Englewood Cliffs, N.J:
Prentice-Hall. pp 89, 97, 102
[4] Chisholm, J.S.R. and Morris, R.M. (1966) Mathematical Methods in Physics. 2nd
Edition. Amsterdam: North-Holland Publishing Company. pp 376, 386
[5] Christie, D.E. (1964) Vector Mechanics. 2nd Edition. New York: McGraw-Hill Book
Company Inc. pp 214
[6] Blyth, M. (2013) Lectures in Mathematical Problem Solving, Mechanics and Mod-
elling. University of East Anglia. High Speed Air Resistance, Chapter 5.3.
[7] Guinness World Records (2014) 8 Roller Coaster Records for National Roller Coaster
Day [Illus.]. Available at: http://www.guinnessworldrecords.com/news/2014/8/
8-roller-coaster-records-for-national-roller-coaster-day-59634
[8] Niles, R. (2013) Ride Review: Full Throttle at Six Flags Magic Mountain [Illus.].
Available at: http://www.themeparkinsider.com/flume/201306/3534/
[9] Marden, D. (2016) Roller Coaster Database. Available at: http://rcdb.com/10948.
htm
[10] PicassoMio (2016) Gallery, Open Listings [Illus.]. Available at: http://www.
picassomio.com/promenades-aeriennes-jardin-baujon-poster-1590161.html
[11] Total Thorpe Park (2016) Features, Dictionary [Illus.]. Available at: http://www.
totalthorpepark.co.uk/features/dictionary.shtml
50

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Rollercoaster Design

  • 1. Roller Coaster Design Connor Hain April 25, 2016
  • 2. Contents 1 Introduction 2 2 Circular Loop 6 2.1 Motion in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Energy Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Air Resistance/Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.1 High Air Resistance . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.2 Low Air Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 G-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Clothoid Loop 16 3.1 Defining the Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Motion Along a General Curve . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Motion Around the Clothoid Loop . . . . . . . . . . . . . . . . . . . . . 22 3.4 General Energy Argument . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.5 G-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.6 Air Resistance/Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4 Circular Helix Loop 29 4.1 Defining the Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Frenet-Serret Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Motion Around the Circular Helix Loop . . . . . . . . . . . . . . . . . . 32 4.4 G-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5 Clothoid Helix Loop 36 5.1 Motion Around the Clothoid Helix Loop . . . . . . . . . . . . . . . . . . 37 5.2 G-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6 Multi-Mass Problem 40 6.1 Two-Mass Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.2 Three-Mass Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.3 N-Mass Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7 Conclusion 47 1
  • 3. 1 Introduction Roller coasters have been the headline attractions at theme parks for many years. Whether they are dominating the skyline or boasting a large number of inversions, they have progressed from being relatively simplistic machines, to becoming highly complex structures, with incredibly complicated control systems. The term ’roller coaster’ is thought to have originated from American thrill rides, from the late nineteenth century. These rides used hundreds of rollers that were mounted to ramps, that allowed an object such as a sled to be ridden down. The passengers in the sled then coasted along the rollers down the ramps, hence the term ’roller coaster’ [1]. However, despite giving these hugely popular machines their name today, these simple American rides had no other impact on the future of roller coasters. The first full circuit roller coaster, Promenades Aeriennes (Aerial Walk), was built early in the nineteenth century (1817) in France (see Figure 1) [1]. Since this achieve- ment, designers have been utilising many different methods in order to create taller, faster and more exciting roller coasters. Today, roller coasters come in a vast array of shapes and sizes. Ranging from a variety of track materials, to different seat configu- rations (e.g. inverted coasters - riders are suspended underneath the track), designers look to create the most thrilling experience whilst keeping the passengers safe. Figure 1: Promenades Aeriennes - first full circuit roller coaster [10]. Throughout the design phase, many complications will arise due to the high number of forces that act on the train, as it makes its way around the circuit. However, before designers can begin to tackle these problems, the method in which the train gathers enough speed to complete the circuit must be determined. There are two main ways in which this is achieved: 2
  • 4. I Lift Hill - The train is pulled up a hill at the beginning of the ride. At the top of the hill, the train is released and runs freely around the track, with only the force due to gravity accelerating it to the required speed to complete the circuit. II Launch System - Many roller coasters now utilise a launch system, which accelerates the train to its top speed in a matter of seconds, so that the circuit can be completed. Once the method of accelerating the train has been determined, designers can then begin to work on the rest of the track layout. There are many factors that influence the layout of a roller coaster, all of which must be taken into consideration. One of the main factors is g-force. G-force creates a problem for designers, because humans can lose consciousness when subjected to a g-force greater than 5g. When a roller coaster is designed, the track layout must compensate for this and should be designed in a specific way, so that riders are subjected to no more than this value. This puts severe restrictions on how a track can be laid out. Early experimentations with vertical loops put riders under as much as 12g [1]. This caused serious injuries, identifying that the shape of the vertical loop needed adjusting. As the train is set into motion, multiple forces act upon it. The force due to gravity powers the roller coaster for some, if not all of its journey around the track. There will also be a reaction force from the track acting on the train. The majority of modern roller coaster trains are designed with special wheel configurations, so that they have wheels on top and underneath tubular tracks [1]. This ensures that the reaction force is never zero, indicating that the train and track always remain in contact with each other. Next, we have an opposing force called the contact force, or friction. This contact force opposes the motion of the train as it moves along the track, consequently slowing the train down by the means of energy conversion; converting kinetic energy to heat energy. Care should be taken to ensure that the contact force is not too great, otherwise significant damage to the track may occur. Drag, D, is also a very important factor to consider when designing a roller coaster. Drag is caused by a body moving through a medium (in this case, air) and pushing the medium out of its path. By Newton’s third law, the medium then exerts equal and opposite forces on the body, consequently slowing the body down. Given that roller coasters often travel at very high velocities, we consider the drag 3
  • 5. equation for an object with high velocity, given by [6] D = −µ|u|u , where µ is the drag coefficient and u is the velocity of the object, with magnitude, u. The drag coefficient is determined by experiment and is affected by the shape of the object moving through the medium. It should also be noted that |D| = −µu2 , showing that the magnitude of the drag force is proportional to the magnitude of the velocity squared. In order to minimise the drag coefficient, fewer and weaker collisions between the air and the car must occur. To achieve this, the front car of the roller coaster train, being the biggest contributor to drag, is designed in particular ways, so that the num- ber/magnitude of the collisions that occur is reduced. The front car usually has a small surface area and is conical in shape, so that fewer air particles collide with the train and the angle at which the collisions occur is reduced, making the collisions weaker. Figure 2: Standard design for the front car of a roller coaster train. All of these forces that act on the train makes modelling a roller coasters motion incredibly complex. It is therefore logical to begin by making some simplifications to the system, before we attempt to perform any calculations. Firstly, we begin by restricting the motion to the x-z plane, with the x-axis being the horizontal and the z-axis being the vertical. At this point we will also ignore the effects of drag and friction. Eventually, we will introduce the effects of drag into the problem, before including the y-axis, so that motion can be studied in three dimensions. The second simplification is that we first model the train as a point mass positioned on top of the track. This prevents dealing with multiple forces that vary at different points throughout a long train at specific times. Also the shape of each car, in particular the front car, will affect the drag, which increases the complexity of the problem. Pro- gressing from a point mass, two point masses can be connected via a light inextensible rod, before adding further masses to the train, so that formulae governing the motion for an n-mass train, where n ∈ N, can be determined. 4
  • 6. Once these simplifications have been made, it is possible to begin considering the motion of a point mass around sections of track; in particular the vertical loop. The vertical loop is one of the most simplistic inversions that can be found on a roller coaster, although the mathematics behind it isn’t. The first vertical loop to be constructed was in Paris in 1846 (Grand Centrifugal Railway) [1]. The ride relied on centripetal forces to keep the car and track in contact with each other. The train would enter the loop at high speeds, ensuring that the reaction force from the track on the train remained positive, when fully inverted at the loop’s highest point. Early vertical loops, including the Grand Centrifugal Railway, were designed so that they were a perfect circle. Beginning in the x-z plane, we will consider a point mass on a circular track and attempt to calculate the magnitude of the initial velocity that must be given to the mass, so that it passes the highest point of the track (the top of the vertical loop). To find a solution to this problem, we need to consider constrained motion/systems. As the point mass is set into motion, we will consider the effects of different forces that are acting on it. In the most simplified case, this is the force due to gravity, along with the normal reaction force. By applying Newton’s second law of motion, we can obtain an equation to calculate the reaction force, along with an equation governing the motion of the point mass along the track. This can be solved using a variety of techniques to find an equation to calculate the magnitude of the initial velocity that is required for the point mass to reach the top of the loop. The equation governing the reaction force is only useful for roller coasters with particular wheel configurations and will not be investigated further. As we progress and add the drag force into the calculations, the equation of motion will be adjusted to account for this extra force. By studying this original problem in more depth, we will discover that this perfect circular shape used in original vertical loop designs, meant that the g-force experienced by the riders was far too great, often resulting in injury. These incredibly high g-forces were caused by the sudden change in the curvature during the transition between the horizontal and circular track. This transition caused a massive onset of centripetal ac- celeration, resulting in the high g-force that is experienced. By finding an equation for g-force and then calculating the turning points of this equation, we will show when g-force takes its maximum value for the circular loop, along with calculating this max- imum value to prove that it is too large. It is because of these high g-forces that were experienced, that adjustments to the shape of a vertical loop were required. The shape of the vertical loop was altered so that it delivered a smooth, comfortable 5
  • 7. ride in 1975 by Werner Stengel [1]. Mathematically, this perfect vertical loop is known as a clothoid loop, similar to the shape of an inverted teardrop and can be modelled by a parametric plot of two integrals, known as the Fresnel integrals. A clothoid loop’s radius of curvature decreases as the train moves from the bottom, to the top of the loop. Along with calculating the magnitude of initial velocity that is required for a point mass to reach the top of a clothoid loop, it will be shown that the transition from a larger radius of curvature at the bottom of the loop, to a smaller radius of curvature at the top, reduces the g-force experienced by the rider to a tolerable amount. For this reason, the clothoid loop became the preferred shape of a vertical loop for roller coaster designers and is still used today in multiple designs. 2 Circular Loop We begin by studying the motion of a point mass around an old fashioned, circular loop. First, consider a point mass, m, which is attached to a circular track of radius a, in the x-z plane (see Figure 3). We will explore two different ways of calculating the magnitude of the mass’ initial velocity, ξ0, that is required for the mass to pass the highest point of the track; the top of the vertical loop. Initially, the only external forces which will be considered are the force due to gravity and the reaction force. We will then extend the problem further by introducing an extra external force, before investigating the problems of using a vertical loop of this shape. First, we define the variables that will be used in the problem. We set θ as the angle between the negative z-axis and the radial unit vector ˆr, R as the reaction force which acts in the −ˆr-direction with magnitude R, ˆθ as the transverse unit vector, g the acceleration due to gravity and ˆi and ˆk as the unit vectors in the x- and z-direction respectively. Here we note that g = −gˆk, where g = 9.81ms−2. 2.1 Motion in Polar Coordinates The problem in question considers a point mass being restricted to moving in a circle. Therefore, we first look to solve the problem using polar coordinates. We define the mass’ position vector as r, with dr dt and d2r dt2 representing the velocity and acceleration respectively. Initially, the mass is positioned at θ = 0, at time t = 0. We want to calculate the magnitude of the initial velocity that is required, for the mass to pass the point θ = π. We take the radius of the circle to be r = a. We apply Newton’s second law to the mass and look for a solution by substituting 6
  • 8. Figure 3: Problem setup using polar coordinates. into this equation. From classical mechanics, we know that the velocity of a point mass in polar coordinates is given by dr dt = dr dt ˆr + r dθ dt ˆθ , (2.1) and the acceleration is given by d2r dt2 = dr dt − r dθ dt 2 ˆr + r d2θ dt2 + 2 dr dt dθ dt ˆθ , (2.2) where r(t) and θ(t) are functions of time, t. Immediately, we can simplify (2.1) and (2.2). Since r is constant, dr dt = d2r dt2 = 0. Therefore, (2.1) reduces to dr dt = a dθ dt ˆθ , (2.3) telling us that the velocity of the mass acts in the ˆθ-direction, tangential to the circle, with magnitude adθ dt . Similarly, from (2.2), we have d2r dt2 = −a dθ dt 2 ˆr + a d2θ dt2 ˆθ . (2.4) Applying Newton’s second law, from (2.4) we have F = ma − dθ dt 2 ˆr + d2θ dt2 ˆθ , (2.5) where m is the mass and F is the total force acting on the mass. 7
  • 9. Next we sum the forces that are acting on the mass to calculate F . This requires us to calculate the component of each force acting in the ˆr- and ˆθ-direction. Resolving in the direction of each unit vector, we find that F = (mg cos θ − R)ˆr − (mg sin θ)ˆθ . (2.6) Here we note that the magnitude of the reaction force in the ˆθ-direction is zero, since R and ˆθ are orthogonal. Then, by substituting (2.6) into (2.5), we obtain ma − dθ dt 2 ˆr + d2θ dt2 ˆθ = (mg cos θ − R)ˆr − (mg sin θ)ˆθ . (2.7) By taking the dot product of this equation with both ˆr and ˆθ, two scalar equations are obtained. We find that ˆr · (2.7) : −ma dθ dt 2 = mg cos θ − R , (2.8) ˆθ · (2.7) : ma d2θ dt2 = −mg sin θ . (2.9) Equation (2.8) can be rearranged so that the reaction force can be determined through- out the mass’ motion. Equation (2.9) is a second order differential equation, which can be solved to find an equation calculating the magnitude of the mass’ velocity. We attempt to solve (2.9) by defining the new variable, ξ = dθ dt , so that differentiating using the chain rule gives d2θ dt2 = d dt dθ dt = d dθ dθ dt dθ dt = dξ dθ ξ . By substituting this into (2.9), we obtain a separable equation, such that aξ dξ dθ = −g sin θ . (2.10) Separating the variables and integrating, gives 1 2 ξ2 = g a cos θ + B , (2.11) where B is a constant of integration. To find this constant of integration, an initial condition is required. At t = 0, the mass is at θ = 0. Therefore, at t = 0, i = ˆθ. Hence, given a velocity, u, with initial magnitude ξ0, from (2.3) we have dr dt = u = ξ0i = a dθ dt ˆθ = aξ ˆθ , 8
  • 10. at t = 0. Therefore, the initial condition is given by θ = 0 and ξ = ξ0 a , at t = 0. This condition will be used throughout Section 2. We apply this condition to equation (2.11) and calculate B, so that after some slight simplification, we arrive at 1 2 ξ2 = g a (cos θ − 1) + ξ2 0 2a2 . (2.12) To calculate ξ0 so that the mass passes the highest point of the loop (θ = π), we introduce a final condition, such that ξ = 0, when θ = π, which will be used throughout Section 2. Applying this condition to equation (2.12) gives an equation that determines a critical value for the magnitude of the initial velocity that should be given to the mass, for it to just reach the top of the loop. Should the mass be given an initial velocity with a higher magnitude than this critical value, the mass will pass the highest point of the loop. After applying the final condition, equation (2.12) reduces to ξ2 0 2a2 = 2g a , which can be rearranged so that we have ξ0 = 2 √ ag . (2.13) Therefore, for a given radius, a, and initial velocity magnitude, ξ0, we have ξ0 > 2 √ ag mass passes θ = π ξ0 < 2 √ ag mass doesn’t pass θ = π (2.14) We can perform a rough check of these inequalities, by calculating ξ0 for a real life example. The tallest vertical loop in the world is the main element of the roller coaster Full Throttle, at Six Flags Magic Mountain, California (see Figure 4) [7]. This roller coaster utilises a launch system, allowing the vertical loop to be the tallest part of the ride. Due to this, the top speed of the roller coaster occurs approximately at the point when the train enters the loop, at θ = 0, to make sure the train successfully completes the inversion. Although the loop is not a perfect circle, we will assume here that it is. The loop has a diameter of 38.75m, therefore its radius is a = 19.375m [7]. Substitut- ing this into (2.13), we find that ξ0 ≈ 61.68mph. This critical value of ξ0 is less than the 70mph top speed of the roller coaster [9]. Since the roller coaster successfully completes the inversion, (2.14) holds, adding confidence that the inequalities are correct. When calculating the initial velocity of a roller coaster, designers want to obtain a highly ac- curate numerical value. Therefore, it should be noted that no rounding has taken place until the final answer is found, so that the error in the calculations is minimised. 9
  • 11. Figure 4: The tallest vertical loop in the world; the main element of the roller coaster Full Throttle, at Six Flags Magic Mountain, California [8]. 2.2 Energy Argument It is possible to calculate (2.13) in an alternate way, by means of an energy argument. From classical mechanics, we know that kinetic energy is given by T = 1 2m|u|2 and the gravitational potential energy is given by V = mgh, where h is the vertical distance between the x-axis and the mass’ position. As the problem does not involve any resisting Figure 5: Resolving diagram for an energy argument. forces, no energy will be lost. Therefore, the system is a conservative system. This allows us to write T + V = B , (2.15) where B is a constant. First, using equation (2.3) we note that |u|2 = a2 dθ dt 2 , giving T = 1 2ma2 dθ dt 2 . Then, resolving from Figure 5, we find that a − h = a cos θ, so that V = mga(1 − cos θ). Substituting T and V into (2.15), gives 1 2 ma2 dθ dt 2 + mga(1 − cos θ) = B . 10
  • 12. Setting ξ = dθ dt and applying the initial condition defined in Section 2.1, we calculate B and arrive at exactly equation (2.12). Equation (2.13) can then be derived from this as before. 2.3 Air Resistance/Drag 2.3.1 High Air Resistance We now look to add the effects of drag, D, into the problem. We begin with the case where the drag coefficient, µ, is large (µ >> 1) and derive an equation for the critical magnitude of the initial velocity, ξ0. An alternate equation can then be found by making adjustments to this equation, for the case when µ is small (µ << 1). As previously discussed in Section 1, when an object has a high velocity, the drag force is given by D = −µ|u|u . (2.16) From Section 2.1, equation (2.3) tells us that u = adθ dt ˆθ, so that |u| = adθ dt . Therefore, we can rewrite D as D = −µa2 dθ dt 2 ˆθ . Next, we apply Newton’s second law to the mass. Starting with equation (2.7), we add the additional drag force to give ma − dθ dt 2 ˆr + d2θ dt2 ˆθ = (mg cos θ − R)ˆr − mg sin θ − µa2 dθ dt 2 ˆθ . Taking the dot product of this equation with ˆθ, we obtain ma d2θ dt2 = −mg sin θ − µa2 dθ dt 2 . Then, by introducing the variable ξ = dθ dt , where d2θ dt2 = ξ dξ dθ , this equation becomes maξ dξ dθ = −mg sin θ − µa2 ξ2 . (2.17) The −µa2ξ2 term makes this equation non-linear. To make this equation linear, we define a new variable w = ξ2, so that differentiating w with respect to θ using the chain rule gives dw dθ = dw dξ dξ dθ = 2ξ dξ dθ . (2.18) Substituting w and (2.18) into (2.17) and rearranging, we then arrive at dw dθ + 2µaw m = − 2g a sin θ . (2.19) 11
  • 13. This equation is of the form dw dθ + p(θ)w = q(θ), therefore, it can be solved by using an integrating factor. After calculating the integrating factor and multiplying (2.19) by this factor, we may write d dθ [we 2µa m θ ] = − 2g a sin θe 2µa m θ . Integrating this equation then gives we 2µa m θ = − 2g a I + A , (2.20) where A is a constant of integration, which can be determined by the initial condition and I is given by I = sin θe 2µa m θ dθ , which must be calculated. To calculate I, we re-write sin θ in terms of the complex exponential. Setting α = 2µa m , the integral becomes I = sin θeαθ dθ = Im(eiθ )eαθ dθ , where i = √ −1 and eiθ = cos θ + i sin θ. Adding the powers of the exponentials and integrating gives I = Im 1 α + i e(α+i)θ . Next, we multiply by α−i α−i and take the imaginary part, so that after some simplification, we arrive at I = 1 α2 + 1 (α sin θ − cos θ)eαθ . Then, by substituting back in for α and substituting I into (2.20), we obtain we 2µa m θ = 2gm a(4µ2a2 + m2) (m cos θ − 2µa sin θ)e 2µa m θ + A . (2.21) Noting that the newly introduced variable was given by w = ξ2, the initial condition introduced in Section 2.1 becomes w(0) = ξ2 0 a2 . Using this condition, we can calculate the constant of integration, A, so that equation (2.21) becomes we 2µa m θ = 2gm a(4µ2a2 + m2) (m cos θ − 2µa sin θ)e 2µa m θ + ξ2 0 a2 − 2gm2 a(4µ2a2 + m2) . (2.22) We now apply the final condition introduced in Section 2.1 to this equation, so that we can determine an equation for the critical magnitude of the initial velocity. This 12
  • 14. final condition requires w = 0, at θ = π. By substituting this condition into (2.22) and rearranging, the critical magnitude equation is given by ξ2 0 = 2gm2a (4µ2a2 + m2) (1 + e 2µa m π ) . (2.23) To check that this equation is consistent with the previous case where drag was neglected, we set µ = 0, giving exactly equation (2.13) as required. 2.3.2 Low Air Resistance Many roller coaster trains are cleverly designed to reduce the drag coefficient as much as possible, so that the drag has minimal effect on the magnitude of the initial velocity that is required for the train to reach the top of the vertical loop. Therefore, we now consider the case where we have very minimal drag (µ << 1) and find an equation for ξ0, which can be used for a train which has been made as aerodynamic as possible. We begin by re-writing equation (2.23) as ξ2 0 = 2gm2a m2(4µ2a2 m2 + 1) (1 + e 2µa m π ) = 2ga (α2 + 1) (1 + eαπ ) , (2.24) where α = 2µa m . We can expand eαπ in Taylor series and (α2 + 1)−1 using a binomial expansion, such that eαπ ≈ 1 + απ + α2π2 2 + O(α3 ) , (α2 + 1)−1 ≈ 1 − α2 + O(α3 ) . By noting that µ << 1 and therefore α << 1, terms of order O(α2) and higher are negligible. Hence, using these expansions, (2.24) becomes ξ2 0 ≈ 2ga(1 + (1 + απ + O(α2 )))(1 − O(α2 )) . Multiplying out the brackets and substituting back in for α gives ξ0 = 2 √ ga 1 + µaπ m 1 2 , where we again use a binomial expansion, neglecting terms of order O(µ2), to give ξ0 ≈ 2 √ ga 1 + µaπ 2m = 2 √ ga + µa 3 2 πg 1 2 m . (2.25) Clearly if µ = 0, this equation for ξ0 is the same as equation (2.13), as required. For a roller coaster train, particularly when loaded with passengers, m will be quite large. By also noting that µ << 1, it can be seen that the extra term appearing in the equation for ξ0 given by (2.25), will have minimal effect on the value of ξ0, showing the benefit of a roller coaster train being aerodynamically designed. 13
  • 15. 2.4 G-Force After calculating an initial velocity for a roller coaster, designers are then faced with another obstacle known as g-force. The g-force acting on an object is defined as the ratio between the force developed from the object accelerating by a cause other than gravity, relative to the force due to gravity on the object [1]. Designers often use g-force to their advantage, by creating moments of weightlessness, along with moments of extreme heaviness for a rider. This is achieved by using specific track shapes, to continually vary the g-force that a rider experiences, throughout the course of the roller coaster. However, despite g-forces creating multiple thrilling sensations for a roller coaster’s riders, they are the biggest limiting factor on the production of more extreme roller coasters. Should riders experience greater than +5g, or less than −5g, serious injury may occur, something designers clearly want to avoid [1]. It is for this reason why the use of a circular loop in a roller coaster’s layout was stopped and a safer, alternative shape was used. By deriving an equation for the g-force, we can identify the problems of using a vertical loop of this shape and calculate where these problems occur. For simplicity, air resistance and friction will be neglected throughout. It is sufficient to show that a circular loop is unsafe by showing that the magnitude of g-force exceeds 5g, regardless of direction. Directly from the definition of g-force, we can write an equation for the magnitude of g-force, G, that is experienced by the mass, at any point on the loop. Recalling the definition, we have that G = m d2r dt2 m|g| = d2r dt2 |g| . (2.26) Immediately it can be seen that g-force is independent of mass. This identifies that two riders of different mass will experience the same g-force as each other. Using equation (2.4), we can write d2r dt2 as d2r dt2 = a dθ dt 4 + d2θ dt2 2 1 2 . (2.27) Recalling that ξ = dθ dt , so that d2θ dt2 = ξ dξ dθ , we can make substitutions for dθ dt and d2θ dt2 using equations (2.10) and (2.12) respectively. Hence, by making these substitutions and then substituting (2.27) into (2.26), we obtain G = a g g2 a2 sin2 θ + ξ2 0 a2 − 2g a + 2g a cos θ 2 1 2 . (2.28) 14
  • 16. Solving dG dθ = 0 will give the turning points of equation (2.28). One of these turning points will be the value of θ at which the magnitude of g-force takes its maximum value. Given that we are only concerned with motion for 0 ≤ θ ≤ π, we will neglect any turning points that are not within this interval. Furthermore, by calculating d2G dθ2 , we can classify the turning points, which in turn allows us to identify what part of the loop generates this maximum value, along with calculating an exact, maximum value of G. We begin by squaring equation (2.28), which, after simplification, gives an equation which can be differentiated much more easily, term-by-term. Differentiating using the chain rule, after further simplification, we arrive at 2G dG dθ = −6 sin θ cos θ − 4ξ2 0 ag sin θ + 8 sin θ . (2.29) For turning points, dG dθ = 0. Setting dG dθ = 0, equation (2.29) simplifies to sin θ −6 cos θ − 4ξ2 0 ag + 8 = 0 . To satisfy this equation, either sin θ = 0, or −6 cos θ − 4ξ2 0 ag + 8 = 0. Using equation (2.13), we make the substitution ξ0 = 2 √ ag, whereby it can be shown that the latter of these equalities requires | cos θ| > 1. This clearly cannot be true. Therefore, the turning points of equation (2.28), for 0 ≤ θ ≤ π, are θ = 0 and θ = π. In order to classify these turning points, we must differentiate equation (2.29) with respect to θ using the product rule, so that we obtain 2G d2G dθ2 + 2 dG dθ 2 = 6 sin2 θ − 6 cos2 θ − 4ξ2 0 ag cos θ + 8 cos θ . (2.30) By definition, when d2G dθ2 > 0 we have a minimum turning point and when d2G dθ2 < 0 we have a maximum turning point. Noting that G > 0 by definition, we substitute θ = 0 and θ = π into equation (2.30) and use equation (2.13) to determine that θ = 0, dG dθ = 0 2Gd2G dθ2 = 2 − 4ξ2 0 ag < 0 → maximum θ = π, dG dθ = 0 2Gd2G dθ2 = 4ξ2 0 ag − 2 > 0 → minimum Given that θ = 0 is the only maximum turning point, the magnitude of g-force is greatest at θ = 0. This is the point of transition between the horizontal and circular track, otherwise known as the entrance to the loop. Since the second half of the vertical loop can be generated by reflection, the g-force for the second half of the loop can also be generated in the same way. This implies that for the entire vertical loop (0 ≤ θ ≤ 2π), we have another maximum turning point at 15
  • 17. θ = 2π; the exit of the vertical loop. When substituted into equation (2.28), θ = 0 and θ = 2π give an equivalent maximum value of G, denoted Gmax. We can calculate Gmax for any circular vertical loop, by substituting θ = 0 (or θ = 2π) into equation (2.28). We reconsider the vertical loop from the Full Throttle roller coaster, described in Section 2.1. Substituting a = 19.375m and ξ0 = 70mph into equation (2.28), we find that Gmax ≈ 5.15g. Since Gmax > 5g, Gmax is too large, showing that this circular shape is not an acceptable shape for a vertical loop. The reason for large g-forces being experienced at the entrance/exit of a circular loop, is due to the large, sudden change in curvature at the point of transition be- tween the horizontal and circular track. This indicates that to make the vertical loop a more comfortable experience for riders, the current large change in curvature at the entrance/exit of the loop must be decreased. This will reduce the g-force experienced, so that for the entirety of the loop, the g-force remains below 5g. This can be achieved by using properties of the clothoid curve. 3 Clothoid Loop The clothoid curve, otherwise known as the Euler/Cornu spiral, has been utilised by roller coaster designers to avoid exerting high g-forces on riders. It has the property that its radius of curvature is inversely proportional to the distance from the center of the spiral, which is equivalent to the curvature varying linearly with arc length [1]. This linear variation property means that a vertical loop modelled in the shape of this curve will have a very minimal change in curvature at the entrance and exit of the loop. Consequently, this should reduce the g-force experienced to a tolerable amount, making the roller coaster safer and more enjoyable for the rider. 3.1 Defining the Curve Before we can consider the motion of a mass around a two-dimensional clothoid loop, we must first find a system of equations that allow us to compute the clothoid curve. This requires some simple definitions, which can then be manipulated to generate the curve. We begin by considering a general curve, C, with arc length, s, in the x-z plane (see Figure 6). A position on this curve is represented by the vector r, where r is represented by two parametric equations, x(s) and z(s), such that r = (x(s), z(s)). We use ˆn and ˆt as the unit normal and unit tangent vectors respectively. 16
  • 18. Figure 6: General curve, C, with defined position and unit vectors. We now define ˆt and ˆn in terms of x(s) and z(s), along with the curvature, κ. At any point on C, the tangential unit vector is given by ˆt = dr ds = dx ds ˆi + dz ds ˆk , (3.1) where ˆi and ˆk are unit vectors in the x- and z-directions respectively. Using the orthog- onal property (ˆt · ˆn = 0), we then define ˆn as ˆn = − dz ds ˆi + dx ds ˆk . (3.2) Finally, the curvature is defined as [4] κ = dˆt ds · ˆn . (3.3) Defining θ as the angle between ˆt and the horizontal x-axis, equation (3.3) can then be rewritten as a relation between θ and the arc length, s. Beginning by taking the dot product of ˆt with ˆi and ˆk, we find from the definition of the dot product that ˆi·ˆt = cos θ and ˆk · ˆt = sin θ. Therefore, from equations (3.1) and (3.2), we can deduce that dx ds = cos θ , dz ds = sin θ , (3.4) ˆt = cos θˆi + sin θˆk , (3.5) ˆn = − sin θˆi + cos θˆk . (3.6) Next we differentiate ˆt with respect to s using the chain rule and note that dˆt dθ = ˆn. This gives dˆt ds = dˆt dθ dθ ds = ˆn dθ ds . (3.7) 17
  • 19. Substituting (3.7) into (3.3), we obtain κ = ˆn · dˆt ds = ˆn · ˆn dθ ds = dθ ds . (3.8) From (3.4) and (3.8), we now have a system of ordinary differential equations. This system can be written as d ds     x z θ     =     cos θ sin θ κ     . (3.9) Solving this system will produce two parametric equations, x(s) and z(s), which describe the clothoid curve. To begin solving (3.9), we recall the definition of the clothoid curve. Since the clothoid’s curvature varies linearly with arc length, we can write κ(s) = αs, where α is a constant. Taking the third equation in system (3.9) and substituting in κ = αs, we use separation of variables to find an equation for θ, such that θ = αs2 2 + θ0 , where θ0 is a constant of integration. Without loss of generality, we take θ0 = 0, so that θ = αs2 2 . We can now substitute θ into the first two equations in system (3.9), before separating the variables and integrating. To integrate the right hand side, we introduce the dummy variable ˆs and integrate with respect to ˆs between zero and s, to give x = s 0 cos αˆs2 2 dˆs , z = s 0 sin αˆs2 2 dˆs . (3.10) Then, by letting ˆs = √ π√ α t, so that dˆs = √ π√ α dt, we obtain x = √ π √ α √ α √ π s 0 cos π 2 t2 dt , (3.11) z = √ π √ α √ α √ π s 0 sin π 2 t2 dt . (3.12) These two integrals are otherwise known as the Fresnel integrals [2]. A parametric plot of these two integrals will produce the clothoid curve required. The Fresnel integrals can also be written as two power series expansions. These expansions converge rapidly for all s (not shown here). For a given value of α, the two series expansions are given by x = √ π √ α ∞ n=0 (−1)n √ α√ π s 4n+1 (π 2 )2n (2n)!(4n + 1) , 18
  • 20. z = √ π √ α ∞ n=0 (−1)n √ α√ π s 4n+3 (π 2 )2n+1 (2n + 1)!(4n + 3) , which can be found in [2]. These can be used to plot the clothoid curve on Matlab. For a vertical loop, the entire clothoid curve is not required. It is important when generating the curve that only the first half of a vertical loop is produced, instead of allowing the curve to become a spiral. The second half of the loop can then be generated by reflection. To ensure that only the first half of a vertical loop is produced, we require a final condition. To find this final condition, we recall the z-integral from (3.10), before the change of variables took place. Differentiating z with respect to ˆs and setting dz dˆs = 0, we calculate the first value of ˆs (excluding ˆs = 0), which satisfies this equation. This value represents the arc length at the point when the clothoid curve becomes horizontal for the first time; the top of the vertical loop. This arc length will be denoted s∗. Calculating dz dˆs and setting dz dˆs = 0, we find that dz dˆs = sin αˆs2 2 = 0 , which first occurs when αˆs2 2 = π. Hence, s∗ = √ 2π√ α . This value will be used as part of the final condition in all calculations throughout this section. Using this condition, it is now possible to compute a variety of vertical loops for different values of α (see Figure 7). By producing a number of curves of varying size, Figure 7: Left: positive clothoid curve. Right: variety of half clothoid loops for different values of α. it can be seen there is a strong relation between every clothoid curve as the value of 19
  • 21. α changes. As shown in Figure 7, the end point of each curve can be connected by a straight line passing through the origin. We can verify this result by substituting s = s∗ = √ 2π√ α into (3.10), giving us two integrals defined as X = √ π √ α √ 2 0 cos π 2 t2 dt , Z = √ π √ α √ 2 0 sin π 2 t2 dt . Then, by defining a shorthand notation for the Fresnel integrals such that C(s) = s 0 cos πt2 2 dt , S(s) = s 0 sin πt2 2 dt , we obtain Z X = √ π√ α S( √ 2) √ π√ α C( √ 2) = S( √ 2) C( √ 2) , which is independent of α, thus verifying the result. 3.2 Motion Along a General Curve Using the Fresnel integrals (given by (3.11) and (3.12)) in conjunction with the final condition for arc length, we can now compute half of a clothoid loop of any height. To allow us to calculate the magnitude of the initial velocity that should be given to a point mass so that it reaches the top of a clothoid loop, we require an equation governing the motion of a mass around the loop. Considering the setup from Figure 6, we will apply Newton’s second law to a point mass and look to find an equation of motion for a mass along a general curve, C. Once calculated, equations describing a specific curve, in this case the clothoid loop, can be applied to this equation of motion, so that we can study the motion of a mass around the clothoid loop. To substitute into Newton’s second law, we must sum the total forces acting on the mass and calculate a formula for d2r dt2 ; the acceleration of the mass. First we calculate d2r dt2 . Differentiating the position vector, r, with respect to t using the chain rule gives dr dt = dr ds ds dt . Since the velocity acts in the tangential direction, dr ds = ˆt. Hence, dr dt = ˆt ds dt . (3.13) 20
  • 22. Differentiating again using the product rule, we obtain d2r dt2 = ˆt d2s d2t + dˆt dt ds dt , which can be re-written by use of the chain rule, such that d2r dt2 = ˆt d2s d2t + dˆt ds ds dt ds dt = ˆt d2s d2t + dˆt ds ds dt 2 . (3.14) Using equations (3.7) and (3.8), we note that dˆt ds = ˆndθ ds = ˆnκ. From this, we can make further simplifications to (3.14), so that d2r dt2 becomes d2r dt2 = ˆt d2s dt2 + ˆnκ ds dt 2 . (3.15) Next, we sum the forces acting on the mass, initially ignoring the effects of drag and friction. Since the reaction force, R, always acts in the ˆn-direction and the force due to gravity acts in the −ˆk-direction, the total force acting on the mass, F , is given by F = Rˆn − mgˆk . Hence, Newton’s second law gives Rˆn − mgˆk = m ˆt d2s dt2 + ˆnκ ds dt 2 . (3.16) Similarly to the circular loop example in Section 2.1, we want to consider the ˆn and ˆt components of this equation. Therefore, by taking the dot product of (3.16) with ˆn and ˆt, we obtain ˆn · (3.16) : R − mgˆk · ˆn = mκ ds dt 2 , (3.17) ˆt · (3.16) : −mgˆk · ˆt = m d2s dt2 . (3.18) Recalling the definitions of ˆt and ˆn from (3.5) and (3.6) respectively, we substitute into equations (3.17) and (3.18), so that after rearrangement, (3.17) gives an equation for the reaction force, which is given by R = mκ ds dt 2 + mg cos θ , (3.19) whilst (3.18) gives us a second order, ordinary differential equation, such that d2s dt2 + g sin θ = 0 . (3.20) By defining θ in terms of arc length for a specific curve, equation (3.20) can now be solved to obtain an equation for the magnitude of the mass’ velocity. It should be noted that by redefining θ in Section 2, we could also solve equation (3.20) for the circular loop, to give us exactly equation (2.13), as before. 21
  • 23. 3.3 Motion Around the Clothoid Loop We now have an equation of motion for a mass along a general curve, given by (3.20). By applying equations used to describe the clothoid curve to this equation of motion, it is possible to calculate an equation which determines the magnitude of the initial velocity that is required for a point mass to reach the top of a loop, as done previously for the circular vertical loop in Section 2.1. Equation (3.20) is initially an equation of three variables. In Section 3.1, we found that θ = αs2 2 for the clothoid. Substituting this into (3.20) gives d2s dt2 + g sin αs2 2 = 0 , (3.21) so that we now have a differential equation in terms of only two variables, s and t. Similarly to the approach in Section 2.1, we introduce the variable ξ = ds dt , so that via the chain rule, d2s dt2 = ξ dξ ds . Substituting this into equation (3.21), we obtain a separable equation, which is given by ξ dξ ds + g sin αs2 2 = 0 . Introducing the dummy variable ˆs, we separate the variables and integrate between zero and s in anticipation of the initial condition, ξ(0) = ξ0, where ξ0 is the magnitude of the initial velocity. This gives 1 2 ξ2 = −g s 0 sin αˆs2 2 dˆs + 1 2 ξ2 0 . (3.22) The integral in this equation can be rewritten by setting ˆs = √ π√ α t, so that dˆs = √ π√ α dt, as seen in Section 3.1. By making this change of variables, equation (3.22) becomes 1 2 ξ2 = −g √ π √ α √ α √ π s 0 sin π 2 t2 dt + 1 2 ξ2 0 . (3.23) Recalling the shorthand notation for the Fresnel integrals, we can rewrite equation (3.23), such that 1 2 ξ2 = −g √ π √ α S √ α √ π s + 1 2 ξ2 0 . (3.24) To obtain an equation for the critical value of ξ0, a final condition is required. We previously calculated in Section 3.1 that at the top of the clothoid loop, s = s∗ = √ 2π√ α . Therefore, the final condition for the mass to just reach the top of the loop is given by ξ(s∗) = 0. Applying this final condition to equation (3.24) gives ξ2 0 = 2g √ π √ α S( √ 2) . (3.25) 22
  • 24. Here we note that S( √ 2) is a known value and can be calculated by the use of Maple. Equation (3.25) can also be derived by using an energy argument. Given that |u| = |ˆtds dt | = ds dt , since |ˆt| = 1 and ds dt > 0, the kinetic energy of the system is given by T = 1 2m ds dt 2 . Then, since the system is conservative, by substituting the z-Fresnel integral given by (3.12) into the equation for gravitational potential energy, we can write 1 2 m ds dt 2 + mg √ π √ α S √ α √ π s = B , where B is a constant. Reintroducing the variable ξ = ds dt and dividing by m gives 1 2 ξ2 + g √ π √ α S √ α √ π s = D , where D is a new constant. By applying the initial condition it can then be shown that D = 1 2ξ2 0. Finally, after applying the final condition to this equation and rearranging, we arrive at equation (3.25), as required. For a given α, it is now possible to determine the minimum magnitude of the initial velocity that must be given to the mass, so that it reaches the top of the vertical loop. By specifying a height, h, for the vertical loop, we substitute s = s∗ and z = h into the z-Fresnel integral given by (3.12), so that we can rearrange to give the required value of α for this height. This can then be used in equation (3.25) to calculate ξ0. It is useful to compare the required initial velocity for a clothoid loop with a circular loop of identical height, so that it can be assessed how using a different loop shape affects the magnitude of the initial velocity that is required. We recall the Full Throttle example used in Section 2.1. The height of the vertical loop is 38.75m, which occurs when s = s∗. By setting z = 38.75m, evaluating the z-Fresnel integral at s = s∗ using Maple and rearranging, we find that α ≈ 1.07 × 10−3. Here we note that this truncated value of α is not used to calculate ξ0. Instead, the exact value is used to minimise the error in the calculation. Substituting the exact values into equation (3.25), gives a critical magnitude of ξ0 ≈ 61.68mph, which is identical to the value calculated for the circular loop. To understand why these values are identical, a general energy argument can be formulated, showing that the equation calculating the magnitude of the initial velocity of a mass along a general curve is independent of curve shape. 3.4 General Energy Argument Consider a general curve with arc length s, connecting two fixed points, P and Q, in the x-z plane (see Figure 8). P is at height z = a and Q is at height z = a + H, where 23
  • 25. H is the vertical distance between P and Q and a ∈ R. Neglecting air resistance and friction, we use simple energy definitions to find an equation to calculate the magnitude of the initial velocity, ξ0, that is required to launch a point mass from P to Q, so that the mass’ velocity is zero as it reaches point Q. Figure 8: General curve between two points, P and Q. Since all resisting forces are neglected, the system is conservative. Hence, we can write T + V = B, where B is constant. In Section 3.3, we found that T = 1 2m ds dt 2 . Thus, noting that V = mgz, we have 1 2 m ds dt 2 + mgz = B . Defining ξ0 as the magnitude of the initial velocity, the initial condition requires that ds dt = ξ0, at z = a. This implies that B = 1 2mξ2 0 +mga. Next we apply the final condition, which requires ds dt = 0, at z = a + H. After some simple rearranging, we are left with ξ2 0 = 2gH , (3.26) which is clearly independent of the path taken as required. This result proves that when neglecting air resistance and friction, for a given height of vertical loop, the shape of the vertical loop has no effect on the value of ξ0. Pro- viding the system is conservative, for loop shapes which are complicated to describe mathematically, equation (3.26) may be used as a useful shortcut to avoid complicated calculations when determining the value of ξ0. 3.5 G-Force Since the clothoid curve’s properties are used primarily to reduce the g-force experienced by riders, it is vital that the g-force experienced when travelling around a clothoid loop 24
  • 26. is calculated, to verify that riders are indeed safe. By recalling the definition for the magnitude of g-force, G, we will apply equations from Section 3.3 to the definition to acquire an equation for G, in terms of arc length, s. Air resistance and friction are neglected throughout. We begin by substituting equation (3.15) into the equation for the magnitude of g-force, which is given by (2.26), so that we have G = 1 g d2s dt2 ˆt + κ ds dt 2 ˆn . (3.27) A substitution can then be made for d2s dt2 using equation (3.20). Similarly, by noting that ds dt = ξ, we can also make a substitution for ds dt 2 using equation (3.24). Therefore, by substituting equations (3.20) and (3.24) into equation (3.27) and acknowledging that for the clothoid, θ = αs2 2 and κ = αs, we obtain G = 1 g −g sin αs2 2 ˆt + αs −2g √ π √ α S √ α √ π s + ξ2 0 ˆn . This is equivalent to writing G = 1 g g2 sin2 αs2 2 + α2 s2 −2g √ π √ α S √ α √ π s + ξ2 0 2 1 2 . (3.28) Solving the equation dG ds = 0 gives the turning points of equation (3.28). Further- more, by calculating d2G ds2 , we can classify these turning points as either minimum or maximum turning points. One of these maximum turning points is the value of s at which the magnitude of g-force is maximum. This maximum magnitude is denoted by Gmax. By substituting each maximum turning point into equation (3.28), the maximum turning point which calculates Gmax can be determined, along with a numerical value for Gmax. For the first half of the clothoid loop, 0 ≤ s ≤ s∗. Therefore, turning points that are not within this interval are neglected. Since the upper bound of the interval is dependent on α, we must first specify a value of α before the turning points can be calculated. It is useful to compare the magnitude of the g-force that is experienced from using a clothoid loop, with a circular loop of identical height. Therefore, we recall the vertical loop from the Full Throttle roller coaster introduced in Section 2.1, for which we previously calculated Gmax ≈ 5.15g for the circular shape loop in Section 2.4. In Section 3.3, it was calculated that α ≈ 1.07 × 10−3 for this vertical loop, which will now be used in the interval, so that the turning points of equation (3.28) can be can be calculated. 25
  • 27. Given the complexity of equation (3.28), we use Maple to calculate the turning points of the equation. By solving dG ds = 0 on Maple, we find that equation (3.28) has three turning points in the range 0 ≤ s ≤ s∗. These are given by s = 0, s = s∗ and s ≈ 40.76m, where s ≈ 40.76m is just over halfway along the first half of the clothoid loop. Furthermore, by calculating d2G ds2 and substituting in the turning points, we find that s = 0 and s = s∗ are minimum turning points and s ≈ 40.76m is a maximum turning point. Since s ≈ 40.76m is the only maximum turning point of equation (3.28), Gmax occurs at this value of s. Recalling that ξ0 = 70mph for the Full Throttle roller coaster, we substitute the non-truncated values of s ≈ 40.76m, α ≈ 1.07 × 10−3 and ξ0 = 70mph into equation (3.28), giving Gmax ≈ 2.50g. Clearly Gmax is substantially lower than 5g, showing that the clothoid is an acceptable shape for a vertical loop. For the clothoid loop, Gmax is less than half the value of Gmax for the circular loop. To visualise how G varies as the mass travels around the two loop shapes, we plot G against s using equations (2.28) and (3.28) (see Figure 9), noting that s = aθ for the circular loop. Figure 9: Graph to show how the magnitude of g-force, G, varies with arc length, s, for a circular loop (red) and a clothoid loop (blue) of identical height (38.75m), for 0 ≤ s ≤ √ 2π√ α . The main problem with the circular loop is the large sudden change in curvature at 26
  • 28. the point of transition between the horizontal and circular track. It can be seen from Figure 9 that the clothoid loop has removed this problem, because G steadily increases from zero, as opposed to the circular loop, where G decreases from its maximum value. It is also clear that Gmax for the circular loop is approximately double Gmax for the clothoid loop, adding confidence that the calculations of Gmax in Sections 2.4 and 3.5 are correct. It should also be noted that s∗ for the circular loop is smaller than s∗ for the clothoid loop. Therefore, in Figure 9, the g-force for the circular loop is plotted for more than the first half of the loop. 3.6 Air Resistance/Drag Now that we have verified that the use of the clothoid curve to model a vertical loop reduces the g-force experienced by riders to a tolerable amount, we look to add an extra external force into the problem; the drag force. The drag force that we add into the problem is given by equation (2.16), such that D = −µ|u|u. Since the velocity, u, points in the tangential direction, we can rewrite D in terms of ˆt. We can then find an equation of motion, so that an equation which determines the magnitude of the initial velocity that is required for a point mass to reach the top of a clothoid loop, with drag included, can be found. From equation (3.13), we know that u = ˆtds dt . Substituting this into (2.16), we have D = −µ ds dt ds dt ˆt = −µ ds dt 2 ˆt , since |ˆt| = 1 and ds dt > 0. Now that we have a relation between D and ˆt, we can substitute into Newton’s second law, giving us equation (3.16) with an additional term on the left hand side, such that Rˆn − mgˆk − µ ds dt 2 ˆt = m ˆt d2s dt2 + ˆnκ ds dt 2 . (3.29) In Section 3.1, we found that θ = αs2 2 for the clothoid. Hence, by taking the dot product of equation (3.29) with ˆt and substituting in for θ, we obtain −mg sin αs2 2 − µ ds dt 2 = m d2s dt2 . (3.30) To solve this equation, we will make use of a substitution. We again introduce the variable ξ = ds dt , so that via the chain rule, d2s dt2 = ξ dξ ds . Substituting this variable into equation (3.30) gives mξ dξ ds = −µξ2 − mg sin αs2 2 . (3.31) 27
  • 29. By making a further substitution of the form w = ξ2, so that via the chain rule, dw ds = 2ξ dξ ds , we can rearrange equation (3.31) to give dw ds + 2µw m = −2g sin αs2 2 . (3.32) This equation is now of the correct form so that it can be solved by using an integrating factor. Letting β = 2µ m , we calculate the integrating factor and multiply equation (3.32) by this factor. Then we may write d ds weβs = −2g sin αs2 2 eβs . (3.33) We now integrate this equation, introducing the limits, zero and s, in anticipation of the initial condition, w(0) = ξ2 0, where ξ0 is the magnitude of the initial velocity, so that we have weβs = −2gI + ξ2 0 , (3.34) where I is given by I = s 0 sin αˆs2 2 eβˆs dˆs , which must be determined. We note that the introduction of the limits has required the use of a dummy variable, ˆs. To calculate I, we first note that sin αˆs2 2 = Im e i αˆs2 2 = Re −ie i αˆs2 2 . Therefore, I can be written as I = Re −i s 0 e i αˆs2 2 +βˆs dˆs . Next, we complete the square for the power of the exponential in the integrand, giving I = Re −ieiβ2 2α s 0 e i α 2 (ˆs+ β iα ) 2 dˆs . Anticipating the appearance of the Fresnel integrals, we write √ π√ 2 t = ˆs + β iα √ α√ 2 , so that dˆs = √ π√ α dt. Substituting this into I, gives I = √ π √ α Re −iei β2 2α √ α √ π s 0 ei(π 2 t2 )dt . It follows that I = √ π √ α Re −iei β2 2α √ α √ π s 0 cos π 2 t2 dt + i √ α √ π s 0 sin π 2 t2 dt , 28
  • 30. where the two integrals appearing in I are the Fresnel integrals defined in Section 3.1, given by (3.11) and (3.12) respectively. Thus, by recalling the shorthand notation for these integrals defined in Section 3.1, we may write I = √ π √ α Re −i cos β2 2α + i sin β2 2α C √ α √ π s + iS √ α √ π s . Taking the real part of this and substituting into equation (3.34), we obtain an equation for w, which is given by w = −2ge−βs √ π √ α cos β2 2α S √ α √ π s + sin β2 2α C √ α √ π s + ξ2 0 . (3.35) We can now apply the final condition to this equation, to obtain an equation which determines the critical value of ξ0. This final condition is given by ξ(s∗) = 0. Hence, in terms of the new variable, w, the final condition becomes w(s∗) = 0. Applying this condition to equation (3.35), we find that the equation for the critical magnitude of the initial velocity is given by ξ2 0 = 2g √ π √ α e −β √ 2π√ α cos β2 2α S( √ 2) + sin β2 2α C( √ 2) . (3.36) By setting µ = 0, so that β = 0, we arrive back at equation (3.25), which is the initial velocity with air resistance neglected, adding confidence that this result is correct. 4 Circular Helix Loop So far, all calculations have been restricted to two dimensions. Although useful, these calculations do not resemble the complete characteristics of a vertical loop. In two dimensions, a vertical loop self intersects, whereas in three dimensions, the track does not intersect. Therefore, to obtain more realistic values for the initial velocity and g- force, we must consider the circular loop and the clothoid loop in three dimensions. We will begin by considering the three-dimensional circular loop, also known as the circular helix loop. 4.1 Defining the Loop We can describe a two-dimensional circular loop in the x-z plane using two parametric equations, which are functions of a variable, t. For a loop of radius a which is centered at (0, a), these two equations are given by x(t) = a sin(t) and z(t) = −a cos(t) + a. In addition to these equations, we can introduce a third parametric equation, which adds 29
  • 31. a linear variance in the y-direction, so that the loop becomes three-dimensional. This linear variance can be written as y(t) = bt, where b represents the pitch of the helix. Restricting 0 ≤ t ≤ 2π, the circular helix loop is therefore described parametrically by x(t) = a sin t , y(t) = bt , z(t) = −a cos t + a . (4.1) For a real life roller coaster, the distance between the entrance and exit of a vertical loop is usually between one and two meters. This distance is kept small so that the loop doesn’t lose its vertical aspect. For this reason, we can therefore assume that old fashioned circular vertical loops also had this property. Given that the distance between the entrance and exit of the circular helix loop, d, is given by d = |y(2π) − y(0)|, we have that b must be in the interval 1 2π ≤ b ≤ 1 π , to ensure that the entrance and exit of the vertical loop are between one and two meters apart. The value of b that is chosen from this interval is largely dependent on the overall size of the loop. We recall the vertical loop element of the Full Throttle roller coaster, introduced in Section 2.1, which has a radius of a = 19.375m. To model this loop as a circular helix loop, we will use b = 1 π , giving a distance of 2m between the entrance and exit of the loop. 4.2 Frenet-Serret Formulas To help us understand motion in three dimensions, we will derive and make use of the Frenet-Serret formulas. Consider a general curve, C, in three-dimensional space with arc length, s (see Figure 10). We define r(s) as the position vector of a point on the curve, ˆt as the tangential unit vector pointing in the direction of motion and ˆn as the unit normal vector. Since ˆt and ˆn are orthogonal unit vectors, they satisfy ˆt · ˆn = 0. Then, we introduce a new unit vector known as the binormal unit vector, ˆb, which is defined as ˆb = ˆt × ˆn [5]. The Frenet-Serret formulas are a set of relations between the three unit vectors, defining how each unit vector varies with arc length. Defining κ as the curvature, we know from (3.7) and (3.8) that dˆt ds = κˆn . (4.2) Next we differentiate ˆb with respect to s, so that we obtain dˆb ds = d ds (ˆt × ˆn) = dˆt ds × ˆn + ˆt × dˆn ds . (4.3) Noting that ˆn × ˆn = 0, we substitute (4.2) into (4.3) to give dˆb ds = ˆt × dˆn ds . 30
  • 32. Figure 10: General curve, C, in three dimensional space, with the position vector for a point on C given by r(s) and the tangential, normal and binormal unit vectors given by ˆt, ˆn and ˆb respectively. From the definition of a vector product, we know from this equation that dˆb ds is orthogonal to ˆt. We also know that ˆb · ˆb = 1. Therefore, dˆb ds · ˆb = 0 = dˆb ds · ˆt . This implies that dˆb ds must be in the direction of ˆn. Thus, defining the magnitude of dˆb ds as −τ, we have dˆb ds = −τ ˆn . (4.4) The variable τ is known as the torsion of a curve. Torsion is a measure of the rate at which a curve is twisting out of its osculating plane at a point [3]. Next, we differentiate the unit vector ˆn, with respect to s. Since ˆb = ˆt× ˆn, we know from use of the right hand rule that ˆn = ˆb × ˆt and ˆt = ˆn × ˆb. Hence, by differentiating ˆn with respect to s and using (4.2) and (4.4), we obtain dˆn ds = d ds (ˆb × ˆt) = dˆb ds × ˆt + ˆb × dˆt ds = −τ ˆn × ˆt + ˆb × κˆn = −κˆt + τˆb . (4.5) The combination of equations (4.2), (4.4) and (4.5) is known as the Frenet-Serret Triad:    dˆt ds = κˆn dˆb ds = −τ ˆn dˆn ds = −κˆt + τˆb (4.6) 31
  • 33. These equations can be found in [5] and [3]. By defining a position vector for a point on a three-dimensional curve, we can use the Frenet-Serret Triad and other geometry definitions to calculate ˆt, ˆn and ˆb, along with κ and τ, which can then be used to derive the governing equations of motion along the curve. 4.3 Motion Around the Circular Helix Loop Using the definition of a circular helix loop, we now define a position vector for a point on the curve. Using the Frenet-Serret Triad and other definitions, we can use this position vector to calculate ˆt, ˆn and ˆb, along with κ and τ, which will allow us to study the motion of a point mass around the loop. We begin by defining r(t) as the position vector of a point along the curve. Using (4.1), we can write r(t) = a sin tˆi + btˆj + (−a cos t + a)ˆk . (4.7) Next we parameterize this position vector, so that it is given in terms of arc length, s. Arc length is given by [3] s = dr dt dt = dx dt 2 + dy dt 2 + dz dt 2 dt . (4.8) Therefore, substituting (4.7) into (4.8) gives s = (a cos t)2 + b2 + (a sin t)2dt = a2 + b2dt = t a2 + b2 . Hence, by defining λ = √ a2 + b2, (4.7) can be written as r(s) = a sin s λ ˆi + bs λ ˆj + (−a cos s λ + a)ˆk . Now that the position vector is in terms of arc length, we can use equation (3.1) to calculate ˆt. Differentiating r with respect to s, we find that ˆt = a λ cos s λ ˆi + b λ ˆj + a λ sin s λ ˆk . (4.9) We now use the first equation of the Frenet-Serret Triad, also given by (4.2), to calculate ˆn and κ. Differentiating ˆt with respect to s, we obtain κ = a λ2 , ˆn = − sin s λ ˆi + cos s λ ˆk . (4.10) Now that we have defined ˆt and ˆn, we can apply them to the definition of the binormal unit vector, so that we can obtain ˆb. Calculating ˆt × ˆn, we have ˆb = b λ cos s λ ˆi − a λ ˆj + b λ sin s λ ˆk . (4.11) 32
  • 34. Finally, we use the second equation of the Frenet-Serret Triad, also given by (4.4), along with equation (4.10), to calculate the torsion. By differentiating ˆb with respect to s, we find that the torsion is given by τ = − b λ2 . This completes the calculation of each component of the Frenet-Serret Triad. A useful check to add confidence that these calculations are correct is to substitute into the final equation of the Triad, also given by (4.5), and check that this equation holds. Ignoring air resistance and friction, we now apply Newton’s second law to the mass and resolve in the direction of the three unit vectors. This will give us a governing equa- tion of motion, which can be solved to find an equation which calculates the magnitude of the initial velocity that is required for the mass to successfully reach the top of a circular helix loop. Given that air resistance and friction are neglected and the loop is circular, by use of the energy argument in Section 3.4, we expect the equation for this magnitude to be identical to the two-dimensional equation, given by (2.13). We begin by summing the forces acting on the mass and use equation (3.15), so that Newton’s second law gives R − mgˆk = m ˆt d2s d2t + ˆnκ ds dt 2 . (4.12) Now that we are working in three-dimensional space, we must take into consideration that the reaction force, R, will also have a component in the ˆb-direction. We write R = Rn ˆn + Rb ˆb , (4.13) where Rn and Rb represent the magnitude of the reaction force in the ˆn- and ˆb-direction respectively. Hence, by substituting (4.13) into equation (4.12) and taking the dot product of this equation with ˆt, ˆn and ˆb, we obtain ˆt · (4.12) : −mg a λ sin s λ = m d2s dt2 , (4.14) ˆn · (4.12) : Rn − mg cos s λ = mκ ds dt 2 , (4.15) ˆb · (4.12) : Rb − mg b λ sin s λ = 0 . (4.16) Equations (4.15) and (4.16) can be rearranged to give an equation to calculate the magnitude of the reaction force in the ˆn- and ˆb-direction respectively. Equation (4.14) is a second order, ordinary differential equation. This can be integrated so that we can 33
  • 35. find an equation for the magnitude of the mass’ velocity at any point on the circular helix loop. To integrate equation (4.14), we introduce the variable ξ = ds dt , so that d2s dt2 = ξ dξ ds . Substituting this into (4.14), gives a separable equation. By separating the variables, integrating by substitution and determining the constant of integration by use of the initial condition, ξ(0) = ξ0, where ξ0 is the magnitude of the initial velocity, we obtain 1 2 ξ2 = ag cos s λ + 1 2 ξ2 0 − ag . (4.17) Now that we are working in three dimensions, we require a new final condition. At the top of the circular helix loop, s = λπ. Therefore, the new final condition is given by ξ(λπ) = 0. Applying this condition to equation (4.17), we find that ξ0 = 2 √ ag , (4.18) which is exactly equation (2.13), as expected. 4.4 G-Force In Section 2.4, we proved that the magnitude of g-force experienced was too large, as a point mass travelled around a two-dimensional circular loop. Although the circular helix loop is also circular in shape, it is useful to see how adding variance in the y-direction to the problem affects the g-force experienced, compared to the two-dimensional problem in Section 2.4. Ignoring air resistance and friction, we substitute equations from Section 4.3 into equation (2.26). This gives us an equation for the magnitude of g-force, G, for which we will calculate the turning points and classify whether these are minimum or maximum turning points. The maximum magnitude of g-force, Gmax, can then be calculated, so that we can determine whether the circular helix is an acceptable shape for a vertical loop. Recalling the equation for the magnitude of g-force given by (2.26), we substitute (4.14) and (4.17) into (2.26) and recall from (4.10) that κ = a λ2 for the circular helix loop, so that G = 1 g − ag λ sin s λ ˆt + a λ2 2ag cos s λ − 1 + ξ2 0 ˆn . This is the same as writing G = 1 g a2g2 λ2 sin2 s λ + a2 λ4 2ag cos s λ − 1 + ξ2 0 2 1 2 . (4.19) 34
  • 36. Next we calculate the turning points of this equation. Following the method from Section 2.4, we square equation (4.19) and differentiate with respect to s to give 2G dG ds = sin s λ cos s λ 2a2 λ3 − 8a4 λ5 + sin s λ 8a4 λ5 − 4a3ξ2 0 gλ5 . (4.20) The turning points are given by dG ds = 0. This occurs when s = nλπ, where n ∈ N, or when cos s λ = 8a4 λ5 − 4a3ξ2 0 gλ5 = 0 . (4.21) Using equation (4.18), we make the substitution ξ0 = 2 √ ag and find that equation (4.21) cannot be satisfied. Therefore, the only turning points are given by s = nλπ. Given that 0 ≤ s ≤ λπ for the first half of the circular helix loop, we neglect turning points that are not within this interval. Hence, the turning points of equation (4.19) that we are interested in are s = 0 and s = λπ. By differentiating equation (4.20), we can classify these turning points. We define two constants: β = 2a2 λ3 − 8a4 λ5 , γ = 8a4 λ5 − 4a3ξ2 0 gλ5 , and then differentiate (4.20) with respect to s, to give 2 dG ds 2 + 2G d2G ds2 = β λ cos2 s λ − β λ sin2 s λ + γ λ cos s λ . By substituting the two turning points into this equation, we find that s = 0, dG ds = 0 2Gd2G ds2 = β λ + γ λ < 0 → maximum s = λπ, dG ds = 0 2Gd2G ds2 = −β λ − γ λ > 0 → minimum Since s = 0 is the only maximum turning point, substituting s = 0 into equation (4.19) will give the maximum magnitude of g-force that is experienced. Recalling the Full Throttle example in Section 2.1, we take a = 19.375m, ξ0 = 70mph and recall from Section 4.3 that λ = √ a2 + b2, where b = 1 π . Substituting these values and s = 0 into equation (4.19), we find that Gmax ≈ 5.15g. Because Gmax > 5g, the circular helix loop is not an acceptable shape for a vertical loop. By comparing this maximum value of Gmax with the value calculated for the two- dimensional circular loop in Section 2.4, we see that the three-dimensional circular loop produces approximately the same value of Gmax. However, since G is dependent upon the pitch, b, this only holds for certain values of b. When b = 0, the vertical loop becomes two-dimensional and the maximum value of g-force will be equal to the value calculated in Section 2.4. For b = 0, the vertical 35
  • 37. loop becomes three-dimensional and we find that as b increases, the g-force experienced decreases. Hence, by increasing b, it would be possible to reduce the g-force so that Gmax < 5g. However, as previously discussed in Section 4.1, when b becomes larger than a certain value (typically b > 1 π for a circular helix loop), the loop loses its vertical aspect and therefore b cannot be increased. Because of this restriction on b, we can conclude that Gmax always has approximately the same value for the two-dimensional and three-dimensional circular loop. By applying properties of the clothoid curve, we can produce a three-dimensional clothoid helix loop, in an attempt to reduce this value to a tolerable amount. 5 Clothoid Helix Loop In two dimensions, the first half of a clothoid loop is defined parametrically by the Fresnel integrals for 0 ≤ s ≤ s∗, with the second half being generated by reflection, as seen in Section 3.1. After redefining these equations, we can introduce a third parametric equation, which adds a linear variance in the y-direction, so that the loop becomes three- dimensional. This three-dimensional loop is known as the clothoid helix loop. We begin by redefining the two Fresnel integrals in terms of the variables t and ζ, such that x(t) = √ π √ α √ α √ π t 0 cos π 2 ζ2 dζ , z(t) = √ π √ α √ α √ π t 0 sin π 2 ζ2 dζ . The linear variance in the y-direction can be written as y(t) = bt, where b is the pitch of the helix. The combination of these three parametric equations, in addition with the condition 0 ≤ t ≤ t∗, defines the first half of the clothoid helix loop. In shorthand notation, these equations are x(t) = √ π √ α C √ α √ π t , y(t) = bt , z(t) = √ π √ α S √ α √ π t . (5.1) For the circular helix in Section 4.1, we set b = 1 π , giving a distance of 2m between the entrance and exit of the loop. So that we are consistent with the distance between the entrance and exit of the two loop shapes, we will choose b so that the clothoid helix loop also has a 2m separation between its entrance and exit. Since the second half of a clothoid helix loop is generated by reflection, t∗ = 2 √ 2π√ α for the full clothoid helix loop. Therefore, we set b = √ α√ 2π . 36
  • 38. 5.1 Motion Around the Clothoid Helix Loop We now consider the motion of a point mass around the clothoid helix loop defined by (5.1). We look to calculate an equation to determine the magnitude of the initial velocity that is required for the mass to just reach the top of the vertical loop. We will achieve this by calculating all components of the Frenet-Serret Triad and applying Newton’s second law to the mass, with air resistance and friction neglected, so that an equation of motion can be found. When solving this equation, we expect to find a critical magnitude equation that is identical to the equation for the two-dimensional clothoid loop given by (3.25), because of the energy argument in Section 3.4. We begin by using (5.1) to define the position vector of a point on the loop, r(t), such that r(t) = √ π √ α C √ α √ π t ˆi + btˆj + √ π √ α S √ α √ π t ˆk . (5.2) Next, we parameterize (5.2), so that r is in terms of arc length, s. Using (4.8), we have s = cos2 αt2 2 + b2 + sin2 αt2 2 1 2 dt = (1 + b2 ) 1 2 dt , giving t = s λ , where λ = (1 + b2). Therefore, (5.2) becomes r(t) = √ π √ α C √ α √ π s λ ˆi + btˆj + √ π √ α S √ α √ π s λ ˆk . (5.3) Now that the position vector is in terms of s, we can use the Frenet-Serret Triad and other geometry definitions to calculate the unit vectors ˆt, ˆn, and ˆb. Along with these unit vectors, we can also calculate the torsion and curvature, given by τ and κ respectively. We start by using equation (3.1) to calculate ˆt. Differentiating (5.3) with respect to s, we find that dˆr ds = ˆt = 1 λ cos αs2 2λ2 ˆi + b λ ˆj + 1 λ sin αs2 2λ2 ˆk . (5.4) Using this equation for ˆt and the first equation from the Frenet-Serret Triad, we can then calculate ˆn and κ. Differentiating (5.4) with respect to s, we obtain κ = αs λ3 , ˆn = − sin αs2 2λ2 ˆi + cos αs2 2λ2 ˆk . (5.5) Next we use the definition of the binormal unit vector to find ˆb. Using (5.4) and (5.5) to calculate ˆt × ˆn, we find that ˆb is given by ˆb = b λ cos αs2 2λ2 ˆi − 1 λ ˆj + b λ sin αs2 2λ2 ˆk . (5.6) 37
  • 39. Finally, we use the second equation of the Frenet-Serret Triad and equation (5.5) to calculate the torsion. By differentiating (5.6) with respect to s, we calculate that τ = − bαs λ3 . This completes the calculation of all components of the Frenet-Serret Triad. We now apply Newton’s second law to the mass and resolve in the direction of the three unit vectors. Ignoring friction and air resistance and recalling the equation for the reaction force, R, in three dimensions given by (4.13), Newton’s second law gives Rn ˆn + Rb ˆb − mgˆk = m ˆt d2s d2t + ˆnκ ds dt 2 . (5.7) Taking the dot product of this equation with ˆt, ˆn and ˆb, we have ˆt · (5.7) : − mg λ sin αs2 2λ2 = m d2s dt2 , (5.8) ˆn · (5.7) : Rn − mg cos αs2 2λ2 = κ ds dt 2 , (5.9) ˆb · (5.7) : Rb − mg cos αs2 2λ2 = 0 . (5.10) By introducing the variable ξ = ds dt , so that via the chain rule, d2s dt2 = ξ dξ ds , we substi- tute ξ into equation (5.8) and attempt to find an equation to determine the magnitude of the initial velocity that is required for a mass to just reach the top of the loop. Substituting ξ into (5.8), we have ξ dξ ds = − g λ sin αs2 2λ2 . Introducing the dummy variable ˆs, we separate the variables and integrate this equation between zero and s in anticipation of the initial condition, ξ(0) = ξ0, where ξ0 is the magnitude of the initial velocity. This gives 1 2 ξ2 = − g λ s 0 sin αˆs2 2λ2 dˆs + 1 2 ξ2 0 . (5.11) Then, in anticipation of the appearance of a Fresnel integral, we write ˆs = λ √ π√ α t, so that dˆs = λ √ π√ α dt. Therefore, equation (5.11) becomes 1 2 ξ2 = −g √ π √ α 1 λ √ α √ π s 0 sin π 2 t2 dt + 1 2 ξ2 0 , (5.12) 38
  • 40. To obtain an equation for the critical value of ξ0, a new final condition is now required. The arc length at the top of the vertical loop, s∗, is given by the first value of s, excluding s = 0, where dz ds = 0. Therefore, given that z = s 0 sin αˆs2 2λ2 dˆs , we have dz dˆs = sin αˆs2 2λ2 = 0 , giving s∗ = λ √ 2π√ α . Thus, the final condition is given by ξ = 0, at s = λ √ 2π√ α . Applying this final condition to equation (5.12), we find that the equation determining the critical value of ξ0 is given by ξ2 0 = 2g √ π √ α S( √ 2) , which is exactly (3.25), as expected. 5.2 G-Force In Section 3.5, we discovered that using a clothoid loop instead of a circular loop reduces the magnitude of g-force experienced when travelling around the loop, to a tolerable amount. Using equations from Section 5.1 where air resistance and friction were ne- glected, we can verify that the three-dimensional clothoid helix loop also reduces the magnitude of g-force, along with comparing how the g-force varies compared to the two-dimensional clothoid loop. The magnitude of g-force, G, is given by equation (3.27), such that G = 1 g d2s dt2 ˆt + κ ds dt 2 ˆn . Given that ξ = ds dt , we substitute equations (5.8) and (5.12) into G, whilst noting that for a clothoid helix loop, κ = αs λ3 , where λ = √ 1 + b2, so that we obtain G = 1 g g λ sin αs2 2λ2 ˆt + αs λ3 −2g √ π √ α S 1 λ √ α √ π s + ξ2 0 ˆn . This is equivalent to writing G = 1 g g2 λ2 sin2 αs2 2λ2 + α2s2 λ6 −2g √ π √ α S 1 λ √ α √ π s + ξ2 0 2 1 2 . (5.13) By differentiating equation (5.13) with respect to s and setting dG ds = 0, we can calculate the turning points. These can then be classified by taking the second derivative of G with respect to s and substituting in the turning points. 39
  • 41. Due to the complexity of equation (5.13), we differentiate using Maple. So that we can compare the three-dimensional clothoid loop with the two-dimensional clothoid loop, we will use the same size vertical loop from the roller coaster Full Throttle for the three-dimensional clothoid loop, as we did for the two-dimensional clothoid loop in Section 3.3. Noting that α is not affected by the problem becoming three-dimensional, we take α ≈ 1.07 × 10−3 and ξ0 = 70mph, with b = √ α√ 2π . Differentiating equation (5.13) once with respect to s and setting dG ds = 0, we find that the turning points of (5.13) are given by s = 0, s = s∗ and s ≈ 40.76m. Differentiating again, we classify s = 0 and s = s∗ as minimum turning points and s ≈ 40.76m as a maximum turning point. Given that we only have one maximum, we substitute the non-truncated value of s ≈ 40.76m into equation (5.13), giving Gmax ≈ 2.50g, which is sufficiently small enough to make the vertical loop safe. Comparing the truncated value of s where the g-force is maximum with the truncated value of s found in Section 3.5 for the two-dimensional clothoid loop, it can be seen that the maximum g-force occurs at approximately the same point on the three-dimensional loop as it does on the two-dimensional loop. Similarly, the truncated values of Gmax are approximately the same in two and three dimensions. This shows that increasing the dimensions of the problem has minimal effect on the g-force and the point where it takes its maximum value. As previously discussed in Section 4.4, since equation (5.13) is dependent upon b, it is possible to reduce Gmax further by increasing b. However, this leads to the vertical loop losing its vertical aspect and because Gmax is already sufficiently small enough, increasing b is not required. 6 Multi-Mass Problem Although some roller coaster trains are made up of one carriage, the majority of roller coasters use a train made up of five to seven carriages. As an extension to the problems we have seen so far, we can consider a multi-mass system, where multiple point masses are joined together, forming a train. Beginning with two connected masses on a two-dimensional circular loop, an equation of motion can be derived for the two masses. Then, by adding further masses to the train, a formula calculating an equation of motion for n masses, where n ∈ N, can be determined. By analysing this equation, the effects of adding extra masses to the train can be discovered. For simplicity, air resistance and friction will be neglected. 40
  • 42. 6.1 Two-Mass Train Consider two point masses, m1 and m2, where m1 = m2, that are connected by a light, inextensible rod, on a two-dimensional circular loop of radius a (see Figure 11). The tension in the rod is given by T , with magnitude T. We define θ as the angle between the negative z-axis and m2, with β being the constant angular separation of m1 and m2. The transverse unit vector is defined as ˆθi and the radial unit vector is defined as ˆri, where i = 1, 2, defining which mass the unit vectors correspond to. Similarly, the position vector of each mass is given by ri and the reaction force for each mass is given by Ri, with magnitude Ri. It should be noted that 0 < β < π. This condition implies that when θ = 0, the first mass in the train has not reached the top of the loop. Figure 11: Problem setup for a two-mass train on a circular loop. We apply Newton’s second law to each mass individually, starting with m1. The total angle at any point between the negative z-axis and m1 is given by θ + β. Recalling the equation for the acceleration of a mass in polar coordinates given by (2.4) and defining ψ = θ + β, we apply Newton’s second law to m1, which gives −ma dψ dt 2 ˆr1 + ma d2ψ dt2 ˆθ1 = −mg sin ψˆθ1 − R1ˆr1 − T ˆx , (6.1) where ˆx is a unit vector pointing in the direction of the tension. Given that r1 = aˆr1 and r2 = aˆr2, we can write ˆx as ˆx = r2 − r1 |r2 − r1| = a(ˆr2 − ˆr1) |ˆr2 − ˆr1| . By calculating |ˆr2 − ˆr1|2 and using the definition of the dot product, it can be shown 41
  • 43. that ˆx = a(ˆr2 − ˆr1) 2 sin β 2 . Next, we apply Newton’s second law to m2, noting that the angle between this mass and the negative z-axis is θ. This gives −ma dθ dt 2 ˆr2 + ma d2θ dt2 ˆθ2 = −mg sin θˆθ2 − R2ˆr2 + T ˆx . (6.2) We wish to solve in the direction of motion for each mass. Therefore, we calculate (6.1) · ˆθ1 and (6.2) · ˆθ2. So that we can calculate ˆx · ˆθi, we convert ˆθi and ˆri into Cartesian coordinates, such that ˆri = sin(φ)ˆi + cos(φ)ˆj , (6.3) ˆθi = − cos(φ)ˆi + sin(φ)ˆj , (6.4) where φ is the total angle between the negative z-axis and the mass in question. Calcu- lating the dot products, we find that (6.1) · ˆθ1 : ma d2ψ dt2 = −mg sin ψ − Ta sin β 2 sin β 2 , (6.5) (6.2) · ˆθ2 : ma d2θ dt2 = −mg sin θ + Ta sin β 2 sin β 2 . (6.6) Since β is constant, d2ψ dt2 = d2θ dt2 . Therefore, by adding equations (6.5) and (6.6) and using a trigonometric identity, we have d2θ dt2 + ˆg2 a sin θ + β 2 = 0 , (6.7) where ˆg2 = g cos β 2 . Clearly if β = 0, we obtain the equation of motion for a single mass train given by (2.9), adding confidence that this equation is correct. Equation (6.7) also tells us that adding an extra mass to the train reduces the effect of gravity that it experiences, where ˆg2 is this new gravitational constant, which is dependent on β. In addition, we note that the angle β 2 is the midpoint of the arc between m1 and m2, indicating that the train behaves as though one mass is positioned at the midpoint between the two masses on the circle. 42
  • 44. 6.2 Three-Mass Train We now add a third mass, m3, to the train and attempt to find an equation of motion in a similar way to the two-mass problem in the previous section. We attach m3 to m2 by a light, inextensible rod, so that θ is now the angle between the negative z-axis and m3 and β is the constant angular separation between each mass, where 0 < 2β < π (see Figure 12). The radial unit vector of each mass is denoted ˆri and the transverse unit vector is denoted ˆθi, where i = 1, 2, 3. Similarly, the position vector of each mass is given by ri and the reaction force is given by Ri, with magnitude Ri. The magnitudes of the tension in the rod connecting m1 and m2 are denoted T12 and T21, where T12 = T21. Likewise, the magnitudes of the tension in the rod connecting m2 and m3 are denoted T23 and T32, where T23 = T32. Figure 12: Problem setup for a three-mass train on a circular loop. We apply the same approach as seen in Section 6.1, starting with m1. Given that the total angle at any point between the negative z-axis and m1 is given by θ + 2β, we define η = θ + 2β, so that Newton’s second law gives −ma dη dt 2 ˆr1 + ma d2η dt2 ˆθ1 = −mg sin ηˆθ1 − R1ˆr1 − T12 ˆx , (6.8) where ˆx is a unit vector pointing in the direction of T21. Similarly, by defining ψ = θ+β and applying Newton’s second law to m2, we obtain −ma dψ dt 2 ˆr2 + ma d2ψ dt2 ˆθ2 = −mg sin ψˆθ2 − R2ˆr2 + T21 ˆx − T23 ˆy , (6.9) 43
  • 45. where ˆy is a unit vector pointing in the direction of T32. Finally, applying Newton’s second law to m3, we have −ma dθ dt 2 ˆr3 + ma d2θ dt2 ˆθ3 = −mg sin θˆθ3 − R3ˆr3 + T32 ˆy . (6.10) Following the method discussed in Section 6.1, it can be shown that ˆx and ˆy are given by ˆx = a(ˆr2 − ˆr1) 2 sin β 2 , ˆy = a(ˆr3 − ˆr2) 2 sin β 2 . Substituting ˆx and ˆy into (6.8), (6.9) and (6.10) and recalling the definitions of ˆri and ˆθi in Cartesian coordinates, which are given by (6.3) and (6.4) respectively, we take the dot product of equations (6.8), (6.9) and (6.10) with their corresponding transverse unit vector. This gives (6.8) · ˆθ1 : ma d2η dt2 = −mg sin η − T12a sin β 2 sin β 2 , (6.11) (6.9) · ˆθ2 : ma d2ψ dt2 = −mg sin ψ + T21a sin β 2 sin β 2 − T23a sin β 2 sin β 2 , (6.12) (6.10) · ˆθ3 : ma d2θ dt2 = −mg sin θ + T32a sin β 2 sin β 2 . (6.13) Since β is a constant, d2η dt2 = d2ψ dt2 = d2θ dt2 . Therefore, by adding equations (6.11), (6.12) and (6.13) and applying a trigonometric identity, we obtain d2θ dt2 + ˆg3 a (sin(θ + β)) = 0 , (6.14) where ˆg3 = g 3 (1 + 2 cos 2β). Again, if β = 0, we obtain the equation of motion for a single mass train given by (2.9). The three-mass train also experiences reduced gravity, where ˆg3 is the new gravitational constant, which is also dependent on β. Since the angular separation of the masses must be within the interval 0 < 2β < π, we have ˆg3 > ˆg2. Thus, the addition of a third mass has further reduced the effects of gravity. To generalise the behaviour of g as the number of masses in the train increases, we can calculate a formula for ˆgn, where n ∈ N is the number of masses in the train. 6.3 N-Mass Train Applying ideas and methods from Sections 6.1 and 6.2, we can easily derive equations of motion for a four, five and six-mass train, which can be used in conjunction with (2.9), 44
  • 46. (6.7) and (6.14), to derive a general formula calculating the equation of motion for an n-mass train, where n ∈ N. This formula is given by d2θ dt2 + ˆgn a sin θ + (n − 1) 2 β = 0 . (6.15) Here ˆgn represents the reduced gravitational constant, θ is the angle between the negative z-axis and mn and β is the constant angular separation of the masses, where 0 < (n − 1)β < π. We find that ˆgn is dependent upon whether n is odd or even, such that n is odd : ˆgn = g n (1 + (n − 1) cos 2β) , (6.16) n is even : ˆgn = 2g n 1 2 n−1 i=0 cos β 2 + iβ . (6.17) For a roller coaster, a typical value of n is given by 1 ≤ n ≤ 7. Using (6.16) and (6.17), we can calculate ˆg1 up to ˆg7 and plot how these vary with β. This allows us to analyse how adding extra masses to the train varies the effects of gravity (see Figure 13 and Figure 14). Figure 13: Graph showing how ˆgn varies with β, for 0 < β < π 6 and 1 ≤ n ≤ 7, where n ∈ N. From Figure 13, we can see that ˆg3 > ˆg4 for 0 < β < π 6 , meaning the addition of an extra mass to the three mass train has not further reduced the gravity experienced. Therefore, given that we found that ˆg3 > ˆg2, it is not possible to draw up any conclusion about the behaviour of ˆgn, for general n. 45
  • 47. Figure 14: Left: graph showing how ˆgn varies with β, where n = 2k, 1 ≤ 2k ≤ 7, and k ∈ N, for 0 < β < π 6 . Right: graph showing how ˆgn varies with β, where n = 2k − 1, 1 ≤ 2k − 1 ≤ 7, and k ∈ N, for 0 < β < π 6 . However, if we consider separate cases for n odd and n even, we can see from Figure 14 that increasing n, decreases the value of ˆgn. From Figure 13 and Figure 14, it is also clear that decreasing the value of β, increases the value of ˆgn. Therefore, given the restriction 0 < (n − 1)β < π, if extra masses are continually added to the train, the angular separation between them must subsequently decrease, so that the effect that each action has on the value of ˆgn should eventually cancel out. We can prove this result by letting n become large and subsequently decreasing β. Let β(n − 1) = φ (φ constant), so that as n → ∞, β → 0. When n is odd, clearly as β → 0, (1 + (n − 1) cos 2β) → n, so that ˆgn → g. For the case when n is even, we let n = 2k, where k ∈ N, so that lim β→0 k−1 i=0 cos β 2 + iβ = k = n 2 . Hence, ˆgn → g. Therefore, by adding more masses to the train and subsequently reduc- ing the angle between them, the reduction in g continually decreases, until there is no reduction whatsoever. Using this result, we note that as n → ∞ and β → 0, equation (6.15) becomes d2θ dt2 + g a sin θ + φ 2 = 0 , where the angle φ 2 is the midpoint of the arc between the first and last mass of the train. This implies that the train will behave as though a single mass is positioned at this 46
  • 48. midpoint. It should be noted that this midpoint signifies the center of mass for all of the combined point masses. Using the formulas for ˆgn, we can compute the reduced gravitational constant for a real life roller coaster. We recall the roller coaster Full Throttle that was introduced in Section 2.1. This roller coaster consists of three cars per train. Assuming that the vertical loop is circular, we use equation (6.16) to calculate ˆg3. Taking into consideration the size of the loop and that β would be small for a real life roller coaster, we approximate the angular separation of the cars as β = π 16, so that ˆg3 ≈ 9.31ms−2. Since g = 9.81ms−2, it is clear from this calculation that for a realistic setup, the reduction in the effects of gravity are minimal. 7 Conclusion From the calculations we have seen, it is evident that the mathematics behind the design of a roller coaster is incredibly complex. After using trial and error in an attempt to design new and exciting track layouts, multiple problems arose. Eventually, designers applied rigorous mathematical techniques to the design process, allowing them to un- derstand what was causing these particular problems with their designs, before seeking solutions to the problems. The vertical loop is an excellent example of this process. The main aim of this project was to replicate the calculations that designers would have made when designing roller coaster vertical loops, for two different loop shapes; the circular loop and the clothoid loop. These calculations included finding the magnitude of the initial velocity that is required for a mass to reach the top of a vertical loop, along with the g-force experienced as the mass travelled around the loop, so that we could identify any problems/benefits of using either loop shape. We began by considering the two-dimensional circular loop (Section 2). The first calculation that was performed was the derivation of an equation linking the height of the vertical loop, to the magnitude of the initial velocity, ξ0, that the train must be given to successfully reach the top of the loop. To simplify this calculation, we used a point mass instead of a full length train. This simplification was used throughout Sections 2-5. We also began by neglecting drag and friction. Using polar coordinates, we found an equation of motion for the point mass on the loop. This equation was then solved and after applying an initial and final condition, we obtained an equation for the critical value of ξ0. This equation was also found using an energy argument. As an extension to this calculation, the drag force was included and 47
  • 49. a similar method was used to find an alternate equation for ξ0. Given the form of the solution, further adjustments using Taylor/binomial expansions were made, allowing an equation for ξ0 to be determined where the drag coefficient was small. After building many roller coasters with circular vertical loops, riders complained about the uncomfortable experiences they were having travelling around these loops. To demonstrate this problem, we mathematically defined the g-force directly from its definition and used equations from Sections 2.1, to find an equation for the magnitude of g-force, G, that is experienced as the mass travelled around the loop. Using values from the roller coaster Full Throttle, it was calculated that this circular shape loop produced a maximum magnitude of g-force which was too high (Gmax ≈ 5.15g), which occurred as the mass entered and exited the loop. This showed why the circular loop is not an acceptable shape for a vertical loop. Next we considered the two-dimensional clothoid loop, which required us to define the loop parametrically. Starting with the basic definition of a clothoid curve, differential geometry was used to derive two integrals known as the Fresnel integrals. Parametrically plotting these integrals produced the clothoid curve. Properties of this curve were then used to create half of a clothoid loop. By using techniques similar to those seen in Section 2.1, an equation determining the critical value of ξ0, with air resistance and friction neglected, was found. By studying the Full Throttle roller coaster, it was shown that the critical value of ξ0 was identical to the value for the circular loop. The reasoning behind this result was then confirmed by an energy argument in Section 3.4. However, the most important calculation was proving that this loop shape does indeed reduce the g-force experienced. For the Full Throttle example, we found that the maximum magnitude of g-force experienced was Gmax ≈ 2.50g, which is slightly less than half of the value for a circular loop of the same height. Thus, it was proven that the clothoid loop was an acceptable loop shape to be used. Next we studied both loop shapes in three dimensions. To understand motion in three-dimensional space, a binormal unit vector was introduced and the Frenet-Serret Triad was derived, which was used to help derive an equation for ξ0. Both loop shapes were found to produce almost identical values of Gmax to their two-dimensional counter- parts. The equations for ξ0 were also identical to the two-dimensional examples, which was verified by the energy argument in Section 3.4. Finally, as an extension to the single mass problem, the behaviour of a train of point masses travelling around a two-dimensional circular loop, with air resistance and friction 48
  • 50. neglected, was investigated. It was discovered that a two-mass train reduced the effects of gravity on the train, which was dependent upon the angular separation of the masses. As more masses were added, we found that the reduction of gravity was dependant on whether the number of cars in the train, n, was even or odd. In both cases, for constant β and 0 < (n − 1)β < π, the reduction in gravity became bigger as the n increased. A formula to calculate the equation of motion for an n-mass train was then determined, which included two formulae for the reduced gravitational constant, ˆgn. Whilst studying the behaviour of ˆgn as n → ∞, so that β → 0, it was seen that ˆgn → g. Vertical loops are one of a variety of inversions that are found on a roller coaster. To further the calculations we have seen, we could mathematically define different inversions and apply these definitions to the equation of motion along a general curve. Then, for each inversion, the g-force and critical value of ξ0 could be calculated by following similar methods that have been seen throughout this project for the vertical loop. These calculations would also require knowledge of other fundamental concepts of roller coaster design, such as banking the track, which is discussed in [1], pages 47-50. An interesting example of an inversion that could be studied is the tilt-incline loop; a vertical loop that is tilted at an angle (see Figure 15). Figure 15: Tilt-incline loop; an inversion found on The Swarm, at Thorpe Park [11]. Inversions, particularly vertical loops, are included in track layouts for hundreds of roller coasters which are operational today. These inversions are used to maximise the thrill and experience of riding a roller coaster. As designers continue to invent taller and faster roller coasters, along with new and exciting inversions, mathematics is vitally important to prove that the passengers will be safe when riding these roller coasters. It is these new inventions that encourage people to visit theme parks every year, highlighting the important role that mathematics plays in the industry and the future construction of all roller coasters. 49
  • 51. References [1] Weisenberg, N. (2013) Coasters 101- An Engineer’s Guide to Roller Coaster Design. 3rd Edition. pp 11-54 [2] Abramowitz, M. and Stegun, I.A. (1972) Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. 10th Edition. New York: Dover Publi- cations. pp 300-301 [3] Widder, D.V. (1961) Advanced Calculus. 2nd Edition. Englewood Cliffs, N.J: Prentice-Hall. pp 89, 97, 102 [4] Chisholm, J.S.R. and Morris, R.M. (1966) Mathematical Methods in Physics. 2nd Edition. Amsterdam: North-Holland Publishing Company. pp 376, 386 [5] Christie, D.E. (1964) Vector Mechanics. 2nd Edition. New York: McGraw-Hill Book Company Inc. pp 214 [6] Blyth, M. (2013) Lectures in Mathematical Problem Solving, Mechanics and Mod- elling. University of East Anglia. High Speed Air Resistance, Chapter 5.3. [7] Guinness World Records (2014) 8 Roller Coaster Records for National Roller Coaster Day [Illus.]. Available at: http://www.guinnessworldrecords.com/news/2014/8/ 8-roller-coaster-records-for-national-roller-coaster-day-59634 [8] Niles, R. (2013) Ride Review: Full Throttle at Six Flags Magic Mountain [Illus.]. Available at: http://www.themeparkinsider.com/flume/201306/3534/ [9] Marden, D. (2016) Roller Coaster Database. Available at: http://rcdb.com/10948. htm [10] PicassoMio (2016) Gallery, Open Listings [Illus.]. Available at: http://www. picassomio.com/promenades-aeriennes-jardin-baujon-poster-1590161.html [11] Total Thorpe Park (2016) Features, Dictionary [Illus.]. Available at: http://www. totalthorpepark.co.uk/features/dictionary.shtml 50