Yet these complicated, surprising movements arise from a remarkably simple geometry. A passenger rides in one of seven cars, each mounted near the edge of its own circular platform but free to pivot about the center. The platforms, in turn, move at a constant speed along an undulating circular track that consists of three identical hills separated by valleys, which tilt the platforms.
The platform movements are perfectly regular, but the cars whirl around independently in an irregular manner. Moreover, there is essentially just one adjustable parameter—the rate at which the platforms move around the track.
When the platforms travel at very low speeds, the cars complete one backward revolution as their platforms go over each hill. In contrast, at high speeds a car gets slammed to its platform’s outer edge and stays locked in that position. In both cases, the motion is predictable.
What happens at intermediate speeds?
To model dynamical systems like the Tilt-A-Whirl, mathematicians, scientists, and engineers use equations that describe how the positions and velocities of a system and its components change over time in response to certain forces.
It's convenient to characterize a system's dynamics by plotting how its position and velocity evolve over time. Each plotted point represents the system’s state of motion at a particular instant, and successive points generate a winding line through an imaginary mathematical space (known as phase space) representing all possible motions. Different starting points generally initiate different curves.
A simple, repeating motion, like the to-and-fro oscillations of a swinging pendulum, appears as a circle or some other closed curve. Such a plot shows that the system cycles through precisely the same state of motion again and again at regular intervals.
More complicated sequences of movements produce tangled paths that wander through phase space, sometimes never forming a closed loop.
Often, it helps to examine such complicated movements not at every moment but at predetermined, regular intervals. In other words, you start with a point representing the system’s initial state, then wait a given time and plot a second point to give the system’s new state, and so on.
In the case of a simple pendulum, selecting an interval equal to the time it takes the pendulum to complete one oscillation produces a plot that consists of a single point. The pendulum is always back in its initial state at every repeated glimpse of its motion.
When the motion is chaotic, however, there is no characteristic period. The resulting plot, known as a Poincaré map, shows points scattered across the plane—like bullets puncturing a sheet of paper. In a sense, the system is continually shifting from one unstable periodic motion to another, giving the appearance of great irregularity.
To describe the Tilt-A-Whirl's dynamics, physicists Bret M. Huggard and Richard L. Kautz developed a mathematical equation that approximates the motion of an idealized Tilt-A-Whirl.
In essence, the movements of an individual car resemble those of a friction-impaired pendulum hanging from a support that is both rotating and being rocked back and forth while the pendulum swings. Solving the equation determines how a Tilt-A-Whirl car would behave under various conditions.
To find out what happens at intermediate Tilt-A-Whirl speeds, Kautz and Huggard plotted a set of points representing the velocity and angle of a car at the beginning of each of 100,000 tilt cycles. They found that the values never repeated themselves but were scattered in a distinctive swirling pattern confined to a portion of the plane.
For these platform velocities, even slight changes in starting point lead to radically different sequences of points. At the same time, it becomes virtually impossible to predict several steps ahead of time precisely what will happen. Such sensitive dependence on initial conditions (Butterfly effect) stands as a hallmark property of chaos.
Hence, what happens to an individual Tilt-A-Whirl car is highly dependent upon the weight of its passengers and where they sit. The resulting jumbled mixture of car rotations never repeats itself exactly, which gives the Tilt-A-Whirl its lively and unpredictable character. Indeed, no two trips are ever likely to produce exactly the same thrills and chills.
At the same time, the mathematical model used by Kautz and Huggard predicts that chaotic motion would occur at a speed close to the 6.5 revolutions per minute at which the ride is normally operated.
Interestingly, Tilt-A-Whirl fanatics know by experience that they can actually take advantage of this sensitivity. They can affect the motion of a car by throwing their weight from side to side at crucial moments, turning cycles with little or no action into thrilling whirls.
"Thus, it would seem that aficionados of the Tilt-A-Whirl have known for some time that chaotic systems can be controlled using small perturbations," Huggard and Kautz observed.
It turns out that Tilt-A-Whirl operators can also take advantage of this sensitivity. Software engineer Dave Boll described his experience one summer running a Tilt-A-Whirl, "which is easily the most entertaining carnival ride to operate."
Why? The operator can actually orchestrate the movement of individual cars. A single lever controls the ride's speed, so an operator can slightly retard or accelerate the ring of platforms at any moment. By applying just the right amount of velocity change at exactly the right time, it's possible to spin a particular car. For example, if a car is currently not spinning, is about to go uphill, and is positioned toward the inside, accelerating the platform will send the car into a very fast spin.
That's what makes the ride so attractive to operate, Boll said.
Of course, any adjustment in speed affects all cars. What happens to particular cars depends on its current spin, its position with respect to the ride's hills and valleys, whether the car is on the inside or outside of its platform, and the velocity change applied by the operator. Very fast spins occur in the same direction as the platform is rotating, and slower spins are in the opposite direction.
Superior spins also provide a bonus for the alert operator. Such whirling inevitably shakes loose coins out of the pockets of passengers—"tips" that can be gathered up after a ride is over and the riders have stepped away!
"There is an art to giving good rides," Boll remarked. "A good operator can sustain a spin on any car, in any direction. After some practice, it is possible to control two cars at once."
"What I usually did was to focus on one particular car, and if I could get another to spin with it, I would try to time it so that both cars maintained their spins as long as possible," he added.
"A walk around an amusement park suggests that several other common rides display chaotic behavior similar to that of the Tilt-A-Whirl," Huggard and Kautz noted. Typically, rides that fit this category have cars that are free to rotate or shift back and forth as they follow a fixed track.
The Tilt-A-Whirl first operated in 1926 at an amusement park in White Bear Lake, Minnesota. Most likely, the ride's inventor, Herbert W. Sellner, discovered its unpredictable dynamics not through mathematical analysis but by building one, trying it out, and making trial-and-error adjustments.
"Ride designers have been fairly adept at finding chaos without appreciating the mathematics underpinning what they’re doing," Kautz noted. The situation is changing, however. To fine-tune the thrills, manufacturers can take advantage of mathematical analyses and computer simulations to help build chaotic motion deliberately into amusement park rides.
Reference:
Kautz, R.L., and B.M. Huggard. 1994. Chaos at the amusement park: Dynamics of the Tilt-A-Whirl. American Journal of Physics 62(January):59.
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