Physics @ Disney
Using an accelerometer, the kinematics of several rides at Walt Disney World in Florida are analyzed.
I recently received an engineering sample of a three-axis MEMs accelerometer chip from Analog Devices - a chip just waiting for the right project! The result is an Accelerometer, and the perfect place to test it was Walt Disney World in Florida. Presented here are the results and complete analysis of the Tower of Terror (one of my personal favourites) as well as data from several other rides.
The actual accelerometer is a small, pocket-sized device, with a large internal memory capacity and built-in graphical LCD display. Data collected can then be downloaded to a PC via a serial link and can be analyzed. The entire design is outlined on the Accelerometer page. The device is seen here in my hand, appropriately with the Twilight Zone Tower Of Terror ride seen in the background. On the rear of the unit are three powerful magnets allowing the device to be attached to ride vehicles - alternately, one can sit on the device to ensure it is attached as best as possible to reduce noise.
The Basic Physics
An accelerometer measures "proper acceleration", which is the physical acceleration an object experiences resulting from either motion or from gravity. We often say that an accelerometer measures "g-force" but this is actually not a force at all - rather what is felt is the force of the earth pushing upwards on the observer preventing them from entering a free fall. Consider Mickey in an elevator (as depicted on the left). While Mickey might "feel" the "pull" of gravity, the real force is exerted upwards by the elevator. A stationary accelerometer will hence read "+1g".
Now, in a free-fall gravity acts on a mass to accelerate it towards the earth. Since nothing pushes upwards on the observer (here, Mickey) feels weightless. The scale upon which Mickey is standing would read zero and an accelerometer in the elevator would read also zero.
So, we often speak of gravity as a "force" but it is really an acceleration - what we feel as gravity is the force that the earth exerts upwards to keep object from entering a free-fall. Regardless of definition, acceleration is often measured in units of "g's" with one "g" defined as an acceleration of 9.8 m/s2.
A similar situation exists in an accelerating vehicle in which the user feels pushed back into the seat. The actual force is in the forward direction and is, in fact, the force of the vehicle pushing on the passenger.
The accelerometer, which uses an Analog Devices ADXL312 accelerometer chip, records acceleration in three axes in the range +/- 12g as signed numbers with 12-bits of resolution. Sitting on a desk, for example, the Z-axis will read, numerically, about 345 (the resolution of the device, then, being about 2.9 milli-G's). In reality, the accelerometer in this device is mounted upside-down due to the necessity to attach leads to the package in which it is supplied (this is then inverted in a spreadsheet although the newer firmware revision does this inside the device.
To record a run, the user simply selects the RECord function and data is stored into a large internal flash memory chip. Data from the chip can then be downloaded serially. The resulting data is in CSV (Comma Separated Value) format with the first number a sequence number, and the resulting three numbers the acceleration values per the following example:
04406,00013,-00582,-00029 04407,00016,-00563,-00017 04408,00021,-00559,-00026 04409,00009,-00561,-00003 04410,00003,-00545,-00012 04411,00011,-00537,-00002 04412,00009,-00528,-00010 04413,00011,-00509,-00028 04414,00009,-00473,-00013 04415,00015,-00447,-00010
In this example, the unit was oriented such that the Y axis is in the vertical (up/down) direction and so the X (column two) and Z (column four) readings hover around zero as expected. There are not many flat surfaces in a ride to mount this device (nor time to look around) so the device was attached to the first steel post found. Inversion of the axis (so that, for example, it reads "+1g" when stationary and not "-1g") is accomplished in the spreadsheet quite easily.
There is a small offset error from the device itself which could have been eliminated by using the self-calibration feature of the ADXL chip (but was not used in this case so small errors must be corrected during analysis). The third column shows an acceleration of approximately -560 or -1.6g but since the unit is actually inverted, the real force is +1.6g (the upwards force of the elevator on the passengers as it rises rapidly. You'll notice as well that all three columns show a noise figure in which the signal varies randomly even with the device sitting still - some of this noise is generated by the shaking of the ride vehicle and some is electrical in nature.
Data from the Twilight Zone Tower of Terror ride, graphed. The horizontal axis of the graph represents sample number (at 25 samples per second), while the vertical axis represents acceleration as a signed number from the chip. All three axes are graphed here. As expected, the X and Z axis hover around zero while the Y axis (in violet here) shows the up/down acceleration of the ride vehicle. The graph is actually inverted so that a negative value for acceleration represents a upward force (obvious from the fact that when the car is not moving, the nominal g-force value is around -345 ... it would be expected that this would be a positive value).
The same data from the accelerometer, inverted to show correct polarity, and with only the Y (up/down) data shown since the other two are irrelevant in this case.
After loading into the ride vehicle, the car ascends at around sample 1200 (the first bump on the graph) - the ride car accelerates upwards, then decelerates to a stop. After a short pause the car accelerates again (around 2400, or about 48 seconds later). The ride car loads into the main drop shaft (the two noisy bumps between 2500 and 3500), and the drops occur. In this case, the ride car encounters a drop on the first entry to the shaft (the first large spike after 3500) resulting in a weightless condition of 0g.
The device provides only acceleration data (in units of g's or, converted, m/s2). Velocity may be found by integrating acceleration with respect to time. Integrating velocity will then result in position.
In the case of this data, a simple numerical method may be used to determine velocity by multiplying acceleration times Δx (which is 40ms in this case where a 25Hz sample rate is employed). A spreadsheet was used to analyze this data as follows:
- The nominal value for one-g is found by averaging several hundred samples while the ride car is stationary. In theory, any one value would work but due to noise in the system, an average is taken. This value is, numerically, approximately 345.
- Raw acceleration data is converted into acceleration in units on m/s2 by dividing the value from the accelerometer by the nominal value for one g (above) and multiplying by 9.8m/s2. This represents the instantaneous acceleration at this time.
- Instantaneous velocity is found by multiplying acceleration by Δx (40ms for the sample rate of 25Hz employed) and adding the last value for velocity (in other words, integrating the velocity). In the spreadsheet this is found by adding the value for velocity in the previous row to the value of acceleration (above) times 0.040. This is in units of m/s.
- Position is found by multiplying velocity by Δx and summing with the previous value of position (again, integrating it).
Position data may then be graphed. With two integrations, it is easy to see how even a small error can result in large errors at the end so each drop sequence was treated separately to avoid cumulative error (seen as "drift" in plots where the ride car remains stationary but the graph continues to slowly climb or fall). The first part of the ride analyzed was the ascent which occurs after guests are loaded into the ride car:
Clearly visible here are two lift sequences. The first brings riders from the loading level (in reality, the second floor of the ride) to a middle position where a twilight-zone-esque video sequence is played. A second lift brings riders to approximately the fifth floor where the ride car travels to the main drop shafts. The data shows, as expected, the load to the main drop shaft occurs at a level of 17m above the loading level (already the second floor).
The main drop sequences are always random on this ride. The first drop occurs immediately after loading into the main shaft. Following that, a sequence of drops, including one large drop, occurs over the next 45 seconds. Finally, the car comes to rest - the graph shows this position to be about 17m below the start position but in reality is would be about one floor lower where guests are unloaded. "Zero" is taken to be the loading position however for absolute position, it should be set to 17m (from the ascent, described previously)
Those who have rode the Tower of Terror will recognize several drop sequences such as the small drop at the top of the shaft (around 12 seconds) in which the car flies to the top of the shaft (allowing guests to see over the entire park), makes a small drop, returning to the top (a "teaser" if you will), followed by a massive drop which pulls you upwards from the seat (the car is actually pulled downward by the ride motors and so is falling faster than a free-fall at that point - verified from the acceleration data). Another feature is the second rise to "almost" the top of the shaft (just below the doors at the top), again with a slight drop and rise, followed by a large drop.
The full data set, along with complete analysis, can be Downloaded Here. This is an XLS spreadsheet from Excel 2007. The Y-axis is corrected for polarity (the fact that the accelerometer is inverted) and graphs are embedded into the spreadsheet.
Unlike the Tower of Terror, the accelerometer cannot be used to determine position - not that it'd be all that interesting, since the ride simply spins in a circle, however the accelerometer cannot sense rotation ... for this a gyroscope is required. And so, the data is presented simple as acceleration data
For the Mission Space ride, the accelerometer was attached to the ferrous metal underside of the console. When the console was pulled forward after loading, it was evident that the accelerometer was almost perfectly horizontal (since the X and Y axes both read zero and the Z axis reads 1.0g upwards). The accelerometer, at this point, is upside-down so that the Z-axis shows a force of +1g (i.e. upwards) not -1g (i.e. downwards) as expected (remember, the accelerometer chip was inverted in the device).
The ride begins at about 8 seconds where the seat reclines so that the rider is almost horizontal (as seen by the decrease in the Z-axis force and the corresponding increase in the X-axis force. The X-axis is now almost horizontal). The rider is left in this position for ten seconds during "launch countdown" after which the ride car is tilted in the axis of the spinning resulting in a large g-force in the X and Z axes. Although the force in each direction appears to be about 1.5 g's, the RMS value (calculated by squaring the g-force in each value, summing these squares, and taking the square root) is very close to 2.0 (the "rated" g-force of this attraction). We then achieve "zero g", simulated by having the X-axis force run from +1.5g to just below 0g by changing the axis of the ride vehicle, again, relative to the spinning axis ... the feeling of going from a large downward force to a small force is perceived as "weightlessness" (and you certainly feel the force pushing you into the seat as being released so the effect is quite convincing). The second stage rocket is engaged at about 67 seconds and another large force is felt by the rider as we travel past the space station towards the moon, followed by "trans-lunar insertion". Another "weightless" sensation during hypersleep is followed by a turbulent ride through asteroids (just before 150 seconds on the graph). Finally, a quick descent to the Martian surface (pulling a large g-force in the process) and a navigating over the mountainous terrain is followed by the "hanging over the cliff" motion at the end (where the X axis rotates so that the car is tipped forward - seen as the X axis going to about -0.3g).
If you wish to correlate the acceleration data to the actual ride visuals, simply go to You Tube and search "Mission Space". By examining the run time of the video and the acceleration data together, you'll get an idea of how it all fits together! The above graph shows where a few scenes from the attraction fit.
This is, of course, the "updated" ride which is generally agreed to be more tame than the original version (which pulled more "g's"). Still, at over 2g's (sustained for over ten seconds), the rider does feel a significant force!
Acceleration from the "Green Team" ride (seen here as raw data, corrected for the fact that the unit was oriented differently than the orange team above) shows similar, but considerably smaller, forces. Since the "green" ride features no spinning, the simulator cars can move only in the Z and X axes and the g-forces are limited to the range -1g to +1g (and at that, -1g would require the rider to be upside-down which does not occur here). So the "weightless" simulation is simply a turn of the ride car from a horizontal position to vertical again.
Rock 'n Roller Coaster
One of the fastest rides at DisneyWorld in Florida, Rock 'n Roller Coaster is a steel roller coaster completely enclosed in a building. It features a unique electromagnetic launch system (a linear motor, really) which propels the car from zero to sixty MPH in 2.8 seconds!
Like most roller coasters, analysis is difficult with acceleration data alone, however the launch system is quite interesting to analyze. First issue: there is no ferrous metal on the ride so the accelerometer was held-down by sitting on it. Since the seats in the ride car are at an angle, a bit of simple physics (in the form of vectors) is required to determine the actual acceleration.
Examining the data from the ride, it can be seen that just before launch (at time t=0) the X axis (front-back) is not zero as expected but shows a nominal value of 0.427G: determined by knowing (from data for the Tower of Terror above) that 1G produces a reading of 344.892. The average reading of the stationary X-axis is 147.5 or 0.427G. Similarly, the Z-axis should read 1G when stationary but reads 0.906G instead.
First, it can be noted that the X and Z accelerations are at right angles to each other so, using the Pythagorean theorem, we obtain a hypotenuse (the actual acceleration in the Z-axis) = √ 0.427 2+0.906 2 = 1.002G as expected.
Simple mathematics can now be used to determine that the accelerometer is inclined at an angle of 25.04 degrees and so, using simple trigonometry, the actual X-axis acceleration is:
The "correction factor" of 1/cos(25.04 degrees),or 1.104, simply means that the actual acceleration in the X-axis is the net acceleration (minus gravity) measured by the inclined accelerometer multiplied by 1.104.
This illustrates how to handle data from the unit even when it is not mounted optimally "flat" to an axis (as it was for the Tower of Terror analysis).
The velocity is then determined by numerically integrating the acceleration as per the same methodology as used with the Tower of Terror analysis. It was found that 2.8 seconds after the acceleration begins, the ride car reaches a velocity of 23.25 m/s (52mph). Errors in the recorded measurements are the likely cause of discrepancy between this and the "accepted" value of 57mph.
This is the most "vicious" roller coaster at Walt Disney World, and riders are subject to serious g-forces both during the forward and backward portions of the ride.
The ride begins at time t=30 seconds (after loading) where the ride car is released. Eight seconds later, after a gentle turn, the car is raised over a short hill and travels to the main lift which begins at 72 seconds as evident from the graph in which the Z-axis force is reduced and the X-axis force is increased indicating a tilt. The lift takes about 50 seconds to reach the top of the mountain, travels downward eventually reaching the "broken track" at 106 seconds. The track behind the car switches (taking about twelve seconds to do this), and travels backwards at high velocity around a sharp bend pulling a whopping FOUR RMS g's (although this is a peak value recorded only for a transient moment - a figure of over 3 g's was recorded for over two seconds during the backwards segment of the ride though). Later, during the forward segment of the ride (the large drop ending with a sharp turn at the bottom), a value of about 2.5 g's (sustained for 1.5 seconds) is felt.