Notation

Before proceeding with any mathematical derivations, an introduction to the notation being used may be helpful. When new key terms are introduced, they will be written in italics.

Parameters are constants that geometrically describe the track and its elements. Each element type requires different parameters to be specified. Variables change along the track. Since the car is constrained to move along the track, this is a single degree of freedom system, with the DOF being the distance along the track. This distance is the independent variable. Functions vary along the track. All functions are dependent on the independent variable. The term quantities will be used to encompass parameters, variables, and functions, essentially referring to all numeric quantities.

Vectors will be represented in boldface, $\mathbf{a}$; while scalars will be represented in plain font, $a$. A unit vector will be represented as a vector with a hat over it, $\ensuremath \mathbf{\widehat{a}}$. In this way, the types of quantities used can easily be distinguished from one another.

Three coordinate systems will be used in this research. All are right-handed, orthogonal systems. See figure 1.3.

Figure 1.3: Coordinate Systems.
The track segment shown is a left-hand upward curve followed by an upward straight section. The curve begins at the origin of the global coordinate system, $(X,Y,Z)$, since it is the first element of the track. The forward vector is labeled $\mathbf{f}$, and the radial vector $\mathbf{r}$. Note that the radial vector has no vertical component. The local coordinate system shown, $(x,y,z)$ starts at the beginning of the straight section.

Global

An inertial coordinate system is established, denoted in capitals as $(X,Y,Z)$, and whose axes are in the $(\ensuremath \mathbf{\widehat{I}},\ensuremath \mathbf{\widehat{J}},\ensuremath \mathbf{\widehat{K}})$ directions. The $Z$ direction is the vertical direction, and is perpendicular to the surface of the earth. The origin of the global coordinate system is placed at the station, where the track begins. The initial track direction is along the $X$ axis. All quantities dealing with the global coordinate system will be expressed in capitals.

Local

An inertial coordinate system is created for each new element that is added. The coordinates are represented in lower case by $(x,y,z)$, with the corresponding axes in the standard $(\ensuremath \mathbf{\widehat{i}},\ensuremath \mathbf{\widehat{j}},\ensuremath \mathbf{\widehat{k}})$ directions. The origin of the local coordinate system is placed at the beginning of the particular track element being described. The local coordinate system is aligned such that the $z$ axis is parallel to the $Z$ axis. The initial track direction is in the $x$-$z$ plane.^1.3

Car-Fixed

A non-inertial coordinate system is created with the origin affixed to the center of gravity of the car. The unit vectors are represented by $(\ensuremath \mathbf{\widehat{f}},\ensuremath \mathbf{\widehat{r}})$^1.4, which represent the forward and radial vectors.^1.5 The forward vector points in the direction the car is moving, while the radial direction points toward the center of curvature.

The distance a car has traveled along the track will be denoted as $S$, while the distance a car has traveled into an element will be denoted as $s$. The total length of an element is denoted by either $l$ or $s_{f}$. Note that the function $s$ is dependent on the independent variable $S$ and the parameters $s_{f_{i}}$ of all previous elements.

$$s = s(S,s_{f_{i}})$$

Figure 1.4: Azimuth and Vertical Angles.
The azimuth is represented by $\theta$, while $\phi$ represents the vertical angle. The vector $\mathbf{f}$ is the forward vector, and $\mathbf{f}_{xy}$ is the projection of the forward vector in the $x$-$y$ plane.

The azimuth angle is denoted as $\theta$, and is defined as the angle between the projection of the forward vector in the $x$-$y$ plane and the $x$ axis. Refer to figure 1.4. The element will be created such that the azimuth angle at the beginning of an element is zero. The azimuth angle at the end of an element is denoted as $\theta_{f}$.

$$\theta = \arctan \left( \frac{dy}{dx} \right)$$

The vertical angle is denoted by $\phi$, and is defined as the angle between the forward direction and the $x$-$y$ plane. Refer again to figure 1.4. The vertical angles at the beginning and end of an element are denoted as $\phi_{0}$ and $\phi_{f}$, respectively.

$$\phi = \arctan \left( \frac{dz}{\sqrt{dx^{2}+dy^{2}}} \right)$$

In dynamic calculations, the derivative with respect to time, as in $\ensuremath \frac{dx}{dt}$, is used frequently and is denoted simply as $\dot{x}$.

^1.3 Note that the initial direction may or may not be along the $x$-axis, due to the possibility of a non-zero vertical angle (described below). However, the initial $y$ value will always be zero.
^1.4 A third vector would be defined as the cross product of $\ensuremath \mathbf{\widehat{f}}$ and $\ensuremath \mathbf{\widehat{r}}$; however, this vector is not needed in the calculations presented here.
^1.5 These correspond to the tangential and normal vectors of the Frenet frame: $(\ensuremath \mathbf{\widehat{t}},\ensuremath \mathbf{\widehat{n}})$[11].