Loop Elements

A loop is used to turn riders upside down. The clothoid loop is an element that can easily be identified on many roller coasters by its ``teardrop'' shape, with a tight radius at the top and a looser radius toward the bottom, which is used is to control the forces on a passenger. Centripetal force is stronger at higher curvature and higher velocities. A clothoid loop was chosen because when higher velocities are encountered (at the bottom of the loop), the curvature is very small. It also conveniently results in continuous curvature, a primary goal of this research.

A loop is located primarily in the $x$-$z$ plane, with a small component in the $y$ direction, to allow the track to shift over enough to pass itself on the way down. A loop can be described with three parameters, described below and depicted in figure 2.5.

Figure 2.5: Loop Element Parameters.
The left figure shows a side view, depicting the ``teardrop'' shape of the $x$-$z$ projection. The range of the clothoid variable, $t$, is also shown. An overhead view is shown in the right figure, illustrating the shift separation.

radius $(r)$

The minimum radius of the loop. This is the constant radius of curvature in the circular region of the $x$-$z$ projection of the loop. ($r > 0$)

circular curve angle $(\delta)$

The angle at the top of the loop with radius $r$. Note that there does not need to be a circular region, although there typically is, as will be discussed below. ( $0 \leq \delta < 2 \pi$)

shift separation $(b)$

The $y$-direction separation between the ends of the loop. ($b > 0$)

The loop should be nearly planar, leaving only enough clearance to allow a car to safely pass through the loop. If a loop with more shift separation is desired, a corkscrew^2.5 may better suit the design requirements.

While the circular region of a loop need not exist, there is good reason to include it. This becomes clear when modeling a train, instead of simply a car. The elevation at the top of the loop is higher with a circular region than it would be without one, since the curvature would continue to increase with a clothoid, pulling the top of the loop downward. As the train navigates the loop, its center of gravity does not follow the path of the loop, but is below the track as the train approaches the top. If no circular region is used, the center of gravity of the train will actually drop as the center of the train approaches the top of the loop, resulting in a speed increase. One of the purposes of using the clothoid in a loop is to control the centripetal forces, putting the highest curvature regions where the lowest speeds are expected to be attained, at the top of the loop. But if the center of gravity of the train drops, the speed will actually increase as the highest curvature region is navigated. Therefore a circular region is used at the top of the loop to keep the center of gravity of the train from dropping as the high curvature region is navigated.

A loop is created in nearly the same manner as a curve. The biggest difference is that the circle/clothoid projection lies in the $x$-$y$ plane for a curve, and in the $x$-$z$ plane for a loop. As with a curve, a loop is created by first determining the coordinates of the projection in this plane from the clothoid parameters. The constant shift angle (as opposed to the vertical angle for a curve) is then calculated from the shift separation, $b$. The shift change ($y$ coordinate) is linear with respect to arclength along the loop.

The clothoid parameters are calculated in the same manner as for a curve. Refer to section 2.1.2. The projected length of a loop in the $x$-$z$ plane is then found from the equations for the arclength of a clothoid and of a circle.

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

Figure 2.6: Loop Shift Angle.
This figure illustrates the relationship between the total length of the loop, the length of the projection in the $x$-$z$ plane, and the shift separation. Note that $\alpha$ is a constant angle throughout the loop, as explained in section 2.1.

The shift angle of a loop, $\alpha$, is comparable to the vertical angle of a curve, $\phi$. Figure 2.6 shows the triangle of figure 2.2 relabeled to apply to a loop. The hypotenuse represents the total arclength of the loop. The lower leg represents the projected length in the $x$-$z$ plane, and the right leg is the shift separation. The angle $\alpha$ is then found from the following.

$$\alpha = \arctan \left( \frac{b}{l_{proj}} \right)$$

The total arclength of the loop is now found from the Pythagorean Theorem.

$$l = \sqrt{l_{proj}^{2} + h^{2}}$$ (2.51)

The initial and final vertical angles of a loop are both zero by definition, which is admittedly a limitation of the definition of a loop. More advanced techniques should ideally be used to allow a loop to begin at a non-zero slope and curvature, but that is beyond the scope of this research.

$$\phi_{0} = \phi_{f} = 0$$ (2.52)

In order to calculate the required geometric functions of a loop, the projected arclength in the $x$-$z$ plane, $s_{proj}$, will be used instead of the arclength along the loop, $s$. The conversion is made with the following, which can be inferred from figure 2.6.

$$s_{proj} = s \cos\alpha$$

^2.5 See section 2.1.4.