Joint Elements

A joint is used to connect the other elements, guaranteeing slope and curvature continuity in the $Z$ direction.^2.8 A joint is used any time the vertical angle at the end of one element is not equal to the vertical angle at the beginning of the next element. Five parameters are needed to specify a joint. See figure 2.10.

Figure 2.10: Joint Element Parameters.
The circular and clothoidal angles are labeled here, along with the minimum radius of the joint.

initial vertical angle $(\phi_{0})$

The vertical angle at the beginning of the joint. $(-\pi/2 < \phi_{0} < \pi/2)$

final vertical angle $(\phi_{f})$

The vertical angle at the end of the joint. $(-\pi/2 < \phi_{f} < \pi/2)$

radius $(r)$

The minimum radius of the joint. This is the constant radius of curvature in the circular region of the joint. $(r > 0)$

entrance clothoid angle $(\gamma_{1})$

Clothoid 1 is the region in which the curvature varies from zero to $1/r$. The vertical angle change experienced by the car as it traverses clothoid 1 is $\gamma_{1}$. The entrance clothoid angle must not be larger than the total included angle of the joint. $(0 \leq \gamma_{1} < \vert\phi_{f} - \phi_{0}\vert - \gamma_{2})$

exit clothoid angle $(\gamma_{2})$

Clothoid 2 is the region in which the curvature varies from $1/r$ back to zero. The vertical angle change experienced by the car as it traverses clothoid 2 is $\gamma_{2}$. The exit clothoid angle must not be larger than the total included angle of the joint. $(0 \leq \gamma_{2} < \vert\phi_{f} - \phi_{0}\vert - \gamma_{1})$

Joints are used to connect the standard elements whenever there is a vertical angle change between the end of one element and the beginning of the next. Since the vertical angles at the ends of the standard elements are specified by those elements, the vertical angles at the ends of the joint are not directly specified. Instead, these angles are chosen to be the same as those of the adjacent elements.

The angle of the circular region is determined from the beginning and ending angles, and from the clothoid angles.

$$\delta = \vert\phi_{f} - \phi_{0}\vert - ( \gamma_{1} + \gamma_{2} )$$

The clothoid parameters are calculated in the same manner as for the other elements. Refer to section 2.1.2. The length of a joint is calculated from the equations for the arclength of a clothoid and of a circle.

$$l = a_{1}t_{1_{f}} + r\delta + a_{2}t_{2_{f}}$$

All elements begin in the $x$-$z$ plane. Since there is no curvature in a joint, $y$ remains zero throughout. The initial and final azimuth angles are both zero.

$$\theta_{f} = 0$$ (2.116)

All sections are first calculated for an upward joint, i. e. $\phi_{f} > \phi_{0}$. If the joint should be downward, the actual joint will be a mirror image of the one calculated. The joint is initially calculated as starting along the $x$ axis, therefore all that must be done to create the mirror image is to negate $z$ after the initial calculation. The joint region is then rotated to properly align it with adjacent track sections. For clothoid 1 and the circular region, this involves a rotation of $\phi_{0}$ radians about the $y$ axis. For clothoid 2 the rotation is $\phi_{f}$ radians. An illustration of the process of creating a joint is shown in figure 2.11.

Figure 2.11: Joint Construction.
This figure shows the same joint of figure 2.10, but with the clothoids and circular region drawn both before and after their $\phi$ rotations, illustrating the joint construction process. Clothoid 1 and the circular region are both initially drawn with the joint starting horizontally. The mirror image is then produced if a downward joint is needed. These regions are then rotated $\phi_{0}$ radians to line up the initial vertical angle of the joint with that of the previous element. Clothoid 2 is initially created ending horizontally. The mirror image is created if necessary. The clothoid is then rotated $\phi_{f}$ radians in order to line it up with the next element.

^2.8 Slope continuity in the $X$-$Y$ plane is guaranteed by the way the elements are connected. See section 2.2. Curvature continuity in the $X$-$Y$ plane is guaranteed by the definition of the elements.