Engineering 101: Stability

The weather is getting warmer, and soon we'll be in the peak of the Swiss construction season. Every year, the winter cold prevents much from happening in the way of civic development. Thus, as soon as the ground thaws, the bulldozers magically spring into action wherever they happened to be last left. A major revamp project is about to begin in front of the main train station here in Winterthur. The central bus depot will be torn down and rebuilt.

Winterthur central bus terminal as seen from the train station

In some ways, it's exciting. I don't mind the old structure, which is not much more than a raised platform surrounded by curb and with a clear plastic roof keeping the rain and snow at bay. However, the modern look they show in the conceptual presentation on the video screens mounted in and about the current place does look appealing. The plan includes a rectangular sloped roof made of aluminum panels with seemingly random sized holes drilled in an array, all jutting out from a central column. The 3D rendering shown morphs into view as a similarly shaped large tree (assumed to be the inspiration) fades. Perhaps even more interesting is the prospect of living here long enough to see a largely familiar landmark removed and replaced by something new, particularly when it will require a good deal of time and effort.

* * * * *

We will have to see how the work all turns out. The Swiss are not necessarily known for short construction projects, but they do a generally admirable job in shuffling things around them. Certain streets simply become one-way, while a bevy of detour signs reroute the poor souls traveling in the opposite direction. Temporary pedestrian walkways are created, some right down the middle of the street, protected by concrete barriers. Crude, but usable stairs are built to join parallel sidewalks which sit at the bottom and top of grassy hills in town, to shorten an otherwise lengthy diversion.

In this case, the solution to a missing central bus hub seems to be a substantial change to the routes. No longer will I be able to hop on and ride a single vehicle all the way to the office. Instead, my familiar #1 bus will morph into a #2 headed back out of town as soon as it gets near the station. A host of changes to the other lines is needed as well.

But all this has little to do with stability, the title of this post. As some of my less "technically-apprised" (i.e. "geeky") readers pointed out, my last post was a bit deeper than they expected . While I am a firm believer that most people underestimate their own mental capabilities, especially when it comes to science or math, I also subscribe to the philosophy that a teacher has a more significant responsibility than the student. Thus, I hope to ease into this subject more gradually than last time, and also to include my favorite of teaching aids: pictures!

I began thinking of stability (which could be defined in multiple ways, but which will be given a clear technical description before the end of this missive) -in the context of my morning commute- the other day as I caught the bus into work. I had just stepped out of the door to my apartment building, only to see my bus driving past, a full 3 minutes earlier than expected. In Switzerland, the public transportation system is quite reliable. This is due both to diligent operators and a well-planned structure. It is the structure that we will investigate here.

"Aw, fiddlesticks!" I said, trudging up the street to the nearest bus stop. "Now I'll be that much later than I normally am to work." And I was, too. Not just because I had to wait for the next bus, but because the next bus was traveling some minutes behind schedule, and would continue to fall further behind all the way to Winterthur main station. The funny thing was: I knew all this would happen, as soon as I saw the first bus flying down the road so far ahead of time. To understand how I knew, you need to understand the concept of stability.

Stability generally refers to a condition of being resistant to change. In engineering, we like also to speak of a state called "equilibrium" where forces are in complete balance. Stability describes how easily that balance is maintained.

Imagine the following: You have a bowl and a marble. The marble is placed in the bowl, and quickly slides down the sides to the center, the lowest spot in the bowl. Now when some force tries to push the marble away from the bottom of the bowl, Earth's gravity and the shape of the bowls sides cause the marble to be forced back to its center spot. "What forces?" you ask. Well, it doesn't really matter, but for the sake of realism let's think of a few possibilities.

This marble has a nice stable home

• A wind blowing across the top of the bowl could easily move the marble to the side.
• A flea's sneeze might give the marble a gentle nudge.
• If the bowl itself is being transported, then some side-to-side movement of the bowl could cause the marble to be displaced relative to it.

These forces are not tremendous; they're certainly not enough to push the marble completely out of the bowl. However, because of the characteristics of the system (the strength of Earth's Gravity, the slope and height of the bowl's sides) the marble will ultimately find itself right back where it started as soon as the external force is removed.

Now, what happens when the bowl is upside-down. That is to say: "What if the system is constructed of an upside-down bowl with a marble on top?" Obviously, the observed behavior will be different, even if the applied outside forces were identical to the first scenario.

This marble will have a much more exciting life, albeit a shorter one

Such a precariously balanced situation is not impossible. However, it generally doesn't last very long since even the small external forces described previously (okay, perhaps not the flea's sneeze) would be sufficient to send the marble hurtling down the outside surface of the bowl. The issue is with the system's stability. Here, the Earth's gravity, and shape of the bowls sides are detrimental to the system's equilibrium. A small movement of the marble to one side, and the system does not provide restoring forces, but rather forces which further drive the system away from the equilibrium point. This system is simply instable.

Recognizing a system's tendency toward instability is the first step in developing countermeasures. With today's microchips and sensors, it is often possible to create electronic controllers which work to keep an instable system operating in a desired position. Here you can see an example of a highly instable system -an upside-down pendulum attached to a small sliding cart - and the computer-controlled motor which slides the cart back and forth to keep the pendulum upright, like a baseball bat standing on your palm. (This is a fairly common university project. We went through similar devices my senior year at Portland State. I chose this video because of the highly professional-looking title cards.)

This kind of strategy is not always practical, however. Sometimes, then, the task falls to the engineer to simply design a more inherently stable system. And this has been attempted, here, in the form of my morning bus commute and with all the buses around this city. The general solution? Short bus routes. For you see, the bus routes are instable, and that was how I knew the next bus was going to be late.

I live in the Northeastern corner of this town, where the bus stops are serviced by only one bus line that runs to the center of town and then the same distance out to the opposite corner. Nevertheless, this bus line contains several buses, so that multiple vehicles are traveling on the route at once. I'm not even sure how many are on the road at the same time. If you examine the bus schedule at the local stop, it simply tells you when a bus is scheduled to arrive throughout the day. During the busiest periods,  this is something like every 6 minutes.

As you ride the bus, you'll notice these small red buttons littered about the walls and railings. They signal the driver to stop at the next station. Thus, if no one riding wants to exit, and there isn't anyone standing at a stop when the bus rolls up, the driver will simply continue on. That the transportation organization is capable of publishing expected bus times, when it's not clear where or how often the bus will stop along it's route is pretty amazing. That they're typically within 1 minute of the actual is even more impressive. Obviously, during Sunday mornings when far fewer people are on the move, the length of time that the route takes is reduced, and all this has been factored into the creation of the posted times. Someone had to do their homework on this for it to work so well.

But what happens if a bus randomly ends up having to make more stops than average? These extra stops take time. And the bus itself becomes more full. Full of people who increase the likelihood of having to make more stops even at stations where no one is waiting to board. In other words, a late bus tends to continue to become even later, classic instability. Okay, so the bus runs a little behind. Big deal. It's what happens next that's interesting. Because there are more than one bus on the road, the lagging of one bus will actually affect the ones following it.

Top: Intended bus spacing. Uniform.
Bottom: Bus spacing in practice. Increasingly non-uniform 

People get used to their bus / tram / train schedules. So a lot of the time, people will be headed to their stop when they know they should. For a good portion of other folks, they get there whenever they get there, especially if it's not during their normal workday commute. If we assume that people show up at bus stops in random fashion, distributed through time in a near-uniform manner, we begin to see how buses affect one another. If the first bus lags behind a small amount, it decreases the span of time between it and the bus right after it. This shorter time means that fewer people than average are expected to be at the stations waiting for the following bus. Fewer people means that the bus will begin to run slightly ahead of schedule. This increases the time between the second bus and the one following it, and the third bus then begins falling behind as well.

The entire bus line as a sequence of evenly spaced buses is an instable system that tends toward alternating fast and slow buses. This alternating sequence is what caused me to recognize that if the bus I just missed was 40% faster than normal, the one following it was quite likely to be substantially slower. After all, it has to go around picking up all the suckers like myself that weren't in time for the first bus.

Taken to the extreme, it can be seen that a system like this left to itself would ultimately end up in a sorry state. After some distance along the route, a trailing bus will actually catch up to the one ahead of it. No one will be riding this trailing bus, since it shows up at the stops literally seconds after the first one. And it will stay like that, with sets of two buses in a row, one packed to the gills and the other empty.

There are two clear methods applied which act to prevent this sort of severe outcome. The first is a mandatory stop for buses which are running too far ahead of schedule. A convenient station about halfway from my apartment to the center of town is used to kill a few moments and let any stragglers sprint to catch the bus, though nobody's ever actually there. I've seen this a few times on lazy weekend mornings, and always feel a bit cheated since I'm actually on the bus wanting to go somewhere. Why should I be forced to wait for imaginary people?

The second method of restoring the desired equilibrium bus spacing is by having more buses than needed for the route. Once a bus gets to the end of the line, whether it's slow or fast, it's replaced on the return route by another bus, leaving at the normal desired time. The shortness of the overall route is then important, to prevent things from becoming too wonky before the vehicles reach the last stop. Imagine how it would be for a fleet of buses running from L.A. to New York, trying to keep an accurate pace to the minute without significant wasted time. Impossible.

With these challenges being satisfactorily met with the current arrangement in a way that allows for reasonably on-time transportation, it remains to be seen how well this change to the bus routes will work. I have faith that an appropriate amount of energy has already been spent ensuring that the proposed plan is the best available, but the proof of the pudding is in the eating, as they say. Come the end of this week, we'll find out whether the route switcheroo reduces system stability to a point where people are significantly affected.