Rollercoaster Loops

We all know the feeling when a car takes a corner too fast and we are thrown outward by the centrifugal force. This effect is deliberately exploited, and accentuated, in designing rollercoasters: rapid twists and turns, surges and plunges thrill the willing riders.

Many modern rollercoasters have vertical loops that take the trains through 360 degree turns with the riders upside-down at the apex. These loops are never circular, for reasons we will explain.

Rollercoaster loops, showing high curvature near the top and lower values lower down.

Circular Vertical Loops

The dynamics of rollercoasters were reviewed in [1], where there are more details than we can present here.

Let’s first examine what happens for a vertical loop. If we ignore frictional losses, the mechanical energy is conserved, with equal values at the top and bottom of the loop:

$\displaystyle \frac{1}{2} M V_T^2 + Mg z_T = \frac{1}{2} M V_B^2 + Mg z_B$

The first term on each side is the kinetic energy, the second is potential energy. The notation is conventional or self-evident. Then

$\displaystyle \frac{1}{2} M V_B^2 = \frac{1}{2} M V_T^2 + 2MgR$

where ${R=\frac{1}{2}(z_T-z_B)}$ is the radius of the loop. Thus, the centrifugal forces at top and bottom differ by ${4g}$ , independent of loop size:

$\displaystyle \frac{V_B^2}{R} = \frac{V_T^2}{R} + 4g$

Suppose now that the speed at the top of the loop is ${V_T=\sqrt{2gR}}$ , resulting in a centrifugal acceleration of ${2g}$ . Since gravity opposes this, the nett acceleration is ${1g}$ upward. At the bottom of the loop, the centrifugal acceleration is ${6g}$ . Adding the acceleration due to gravity, the total ${g}$ -force downward is ${7g}$ . This is more than most people could or would tolerate.

Clearly, circular loops are unsuitable for rollorcoasters. Many other loop shapes have been tried. It is easy to design loops for which the centrifugal force is constant. Loops with constant ${g}$ -force are another possibility, though they may not be so exciting for riders. We look at a third possibility, using clothoids.

Clothoid loops

Clothoids are segments of a spiral curve first studied by the great Swiss mathematician Leonhard Euler in 1744. Later, and independently, the spiral was re-discovered by the French physicist Marie Alfred Cornu when he studied the diffraction, or spreading of light passing through a narrow slit, and it is now known as the Cornu spiral.

A clothoid has curvature that increases linearly with distance along the curve. Suppose ${s}$ gives arc-length along the curve, and ${\theta}$ is the angle between the ${x}$ -axis and the tangent to the curve. Then the curvature ${\kappa}$ is the rate of change of ${\theta}$ with ${s}$ , so we have

$\displaystyle d\theta = \kappa\,ds \qquad dx = \cos\theta\,ds \qquad dy = \sin\theta\,ds$

Now suppose the curvature varies linearly with arc-length, ${\kappa = cs}$ . Then, with obvious choice of initial values, ${\theta = \frac{1}{2}cs^2}$ and ${x}$ and ${y}$ are given by

$\displaystyle x = \int \cos\frac{cs^2}{2}\,ds \qquad y= \int \sin\frac{cs^2}{2}\,ds$

These are standard integrals called Fresnel integrals. Plotting ${y}$ against ${x}$ we get the Cornu spiral, shown in the figure below.

Cornu spiral. Curvature increases linearly with arc-length.

We have seen that a circular loop might have a centrifugal force of ${6g}$ or ${7g}$ at its lowest point. If the circular arc is linked directly to a horizontal section, the onset of this force will be abrupt, and very uncomfortable. To avoid sudden changes, clothoids are used to ensure a gradual increase in centrifugal force. The use of clothoids for rollercoaster design was pioneered in 1975 by engineer Werner Stengel, founder of Stengel Engineering (Ingenieur Büro Stengel GmbH).

Vertical loop on the Shockwave coaster at Six Flags over Texas [image Wikimedia Commons].

The figure above shows a loop on a rollercoaster. (See also the figure at the top of this post). It is clear that the curvature is large at the top of the loops and smaller lower down. The centrifugal force is proportional to the curvature, so this ensures that the ${g}$ -force does not reach an unacceptable level.

Sources

${\bullet}$ [1] Pendrill, Ann-Marie, 2005: Rollercoaster loop shapes. Phys. Education, 40, 517-521. http://physics.gu.se/LISEBERG/eng/pe5601.pdf

* * * * * *

Peter Lynch’s book about walking around the coastal counties of Ireland is now available as an ebook (at a very low price). Order now from amazon. For more information and photographs go to http://www.ramblingroundireland.com/

ThatsMaths