No, not pie...we're talking numbers! Pi! π!

Much of the calculations and scientific analysis for engineering applications can be done using approximations and valid assumptions to simplify an otherwise grossly complicated problem. If you were to take into consideration every single force, large or small, that could possibly interact with the system that you’re investigating, then your analysis work would become overblown to mythic proportions, and you might end up running away in utter confusion. Typically in engineering analysis, you neglect anything that is insignificant compared to the “big picture.” For example, if you’re analyzing how a car reacts to sliding on ice, you would usually assume kinetic friction is negligible, even though it’s there on a very small scale compared to the car’s motion on the ice.

One common approximation that engineers use in their numerical analysis is for π (we have a post from a few weeks ago about the use of Greek letters in engineering). As you might already know, you cannot be exact with the value of pi because it is an irrational number. There are, of course, several different ways to approximate the value of pi. So instead of using 3.141592653589793238462643383279502884197169399… (continued indefinitely), we use a simple 3.14 for π when doing calculations. Most people (not just engineers) would make this approximation if they were faced with a problem involving π. Now, you may know some people who can recite several hundred digits of π right off the top of their heads, and while it is a pretty cool skill to have and does have the potential to amaze the heck out of people, there don’t seem to be many practical applications associated with it. A simple 3.14 is good enough for most problems in engineering. Of course, if you’re looking for a more exact solution to a problem, then you’ll need to tighten your approximations to allow for more accurate solutions.

For your edification, pi to as many digits as you may care to want; 1000 digits for example: http://www77.wolframalpha.com/input/?i=N[pi,1000]. Just change the 1000 to represent however many you want.

(Image from Wikipedia)

Hypermiling Bumper Sticker

***Disclaimer: Real hypermiling is pretty dangerous and some aspects of it are illegal, so don’t break the law. Also, I do not really care about the status of the environment; I only hypermile for fun.

These days it seems that fuel economy is the selling point of every new car. Most people just look at these overused numbers (like, 32 mpg Highway!) and don’t really understand what it means. If they are comparing two similar vehicles, though, then value might actually come down to the best fuel-economy (most of the time, vehicle comparisons are not fair). But chances are that person will not actually achieve the EPA-rated numbers for numerous reasons. To hypermile is to drive in a way that minimizes fuel consumption, and there are many ways to try it. A lot of American drivers like to accelerate quickly for no apparent reason. If you are driving in the city, you should drag out the 0-30 acceleration as long as you can (but it’ll probably depends on the general attitude of other drivers in your city).

During my week of total dedication to hypermiling, I don’t think I pressed the gas pedal past about 10%, which is tough on the mind if you live for turbocharged acceleration as I do. If you see a light going yellow in your path, instantly let off the gas. If you know the light pattern well, shut off your engine if you’ll be idling at a red light for a little while. (I’ve found this helps a lot at some rural lights that seem to never change.) Also try not to slow down as much into turns, but keep it safe. Carrying speed, i.e. momentum, is the best way to achieve better mileage. Run your engine at lower RPMs, if possible, just not if you are breaking in a new car. I don’t think I used anything past 2500 and I shifted to the top gear as soon as I could. My friend was telling me that he hardly sees anything more than 25 mpg out of his 2.5 Subaru engine, which is unusual for someone that cares about mileage, but when I rode around with him I discovered that he ran the car at about 3500 RPM on normal roads when he could be shifting. Since the engine spins only about 72% as much at 2500 than at 3500, this can be a significant problem. I also like to coast in neutral up to stop lights and skip shifts (3^rd to 6^th) to keep engine speeds down.

Hypermiling is a fun challenge and I find myself trying to beat my record each time I make the 100 mile (mostly 45-55 mph roads) commute back home throughout the summer. With a record of 38.8 mpg out of my 2.0T (and without angry drivers or Amish buggies stuck behind me), I’m somewhat above the rated value of 21 city/31 highway for my model. Almost any car can perform well above its rating if driven correctly. The regular checks such as keeping your tires properly inflated and removing roof racks for extended trips will also help out.

Good Ways to Get Started

If you’re looking to buy a new car, try something small, a family of four should have no problem fitting in a mid-size sedan orwagon. Try to find engines with small displacements that put out as much power as larger engines. Used diesels are probably the best you can get these days from a miles per dollar standpoint. Older VW diesel models are known to get 50+ mpg without any hypermiling techniques, and when coupled with some basic strategies, they can deliver better numbers than almost any hybrid for a fraction of the cost.

Outside of my studies (more useful and more common in everyday life), I have tried to develop a sense for these “foreign” things. I started off with the 24-hour system about 5 years ago. Now, it’s second-nature. I have since moved on to getting a feel for the Celsius temperature scale.

The obvious thing to do would probably be to memorize a few formulas for the conversion of what you know to what you want to know (i.e. number of miles to number of kilometers). The same applies to the all-too-familiar equation of Fahrenheit to Celsius, and vice versa. Here’s what I’ve discovered (note: some math may be required).

C to F Conversion

F to C Conversion

1. 24-Hours

We all know that the 24-hour clock is nothing more than a means to get rid of the AM/PM suffix at the end of our times. It is a cleaner way to represent time and it completely avoids ambiguity (maybe except for 00:00 and 24:00, which represent the same time). Say the time is 21:47. To convert that craziness to something we know, what would we do? Of course we would take 21:47 and subtract 12:00. Depending on how proficient you are at finding the difference of times, you’ll quickly arrive at 9:47 pm (the pm comes from the fact that it is past 12:00 noon).

Someone once told me that rather than doing the subtraction, or going through the motions of doing mental math, simply take the “21″ and decrement it by 2 (which comes from subtracting the “12″). Two less than 21 is 19 which “rounds down” to 9.

You can argue that it is the same as subtracting. Well, it is, to be sure. But this decrementing by two is far easier for me than pretending I can do subtraction in my head. So, 18:07 is just 6:07 pm.

2. Celsius Temperature Scale

Recently, I was toying around with the Weather widget on my iPhone. I converted it to Celsius and I wanted to see how long it would take for me to get frustrated and switch it back to degrees F. Well, in the practical range of moderate temperature, we can simply approximate the temperature formula. 9/5 becomes about 2, and 32 is about 30. The formula goes from [9/5*C + 32] to [2*C + 30], which is far easier to work with, I’d say.

Let’s assume the temperature is 20 degrees C. Doing some thinking, we take 20, multiply it by 2, and add 30. It is about 70 degrees F. (If we use the proper formula we get 68 degrees F.) Close enough!

Be careful though. This approximating becomes terribly useless as you get further from the “linear range” of -10 to +30 degrees C. Most of the time, it wouldn’t matter if you’re just considering the weather, but the approximation deviates more and more from actuality. For small degrees C, it would underapproximate, and for larger degrees C, it would overapproximate. Just keep that in mind.

Reflection on a Pond has a One-to-One Correspondence; It Doesn't Flip Horizontally

So we were talking about plane mirrors the other day. What makes mirrors so special that they apparently only only reflect left and right and not up and down? After some discussion, we jokingly decided that  it was because of gravity, because one way or another, we can usually blame gravity for just about anything.

We’re kidding.

Some people say it’s the eyes or brain that interprets a reflection from a mirror the “proper” way we see it. This assertion cannot be true. While it is true our eyes gathers light and inverts them in our eyes (like with a concave lens, for example), our brain interprets it inverted as well, which makes the image “upright” in our minds. The eye is simply a light sensor; they way it gathers light is not a function of what the light source is (it doesn’t change the way it takes in light depending on what you’re looking at).

Actually, plane mirrors are simpler than most people probably make it out to be. Let’s try this. On a piece of paper, write the word “TEST”. Now show it to the mirror to read its reflection. As we expect, the letters are reflected horizontally. Now, write the word “TEST” on a semi-translucent piece of paper. Hold up this paper up to the mirror, but this time have the word facing you. Now, we see that you can also read “TEST” the proper way in the mirror as well; it’s normal!

From this test, we can see that mirrors reflect left to the left and right to the right. Now, how does this work? Consider meeting someone face-to-face and shaking hands. In a right-handed society, you’ll extend your right hand and the other person will extend his right hand. To you though, he appears to be extending from his “left” side. The same idea applies to plane mirrors. The only reason a mirror appears to reflect horizontally is because you actually turned to look at it (as if walking up to meet your virtual self). In other words, the reason why objects (or letters) look reversed in a mirror is because you are presenting them to the mirror already in reverse.

Try it! It’s cool.

Wheel Imbalance Can Easily Be Detected When Driving on the Highway

Have you ever driven a car that vibrated uncontrollably at certain driving speeds, especially on the highway? If you have, you probably noticed that these large vibrations occur when you drive at one specific speed, as shown on your speedometer, and the vibrations die out when you travel a little lower or higher than that specific speed. Personally, I’ve driven a car that undergoes unusually large vibrations when I’m driving it at around 70 miles per hour. This phenomenon arises from an imbalanced rotor.

Rotors are objects that rotate, like wheels on a car. An imbalanced rotor is one whose center of mass is not in line with its axis of rotation. For a wheel, the axis of rotation would be the axle of the car. Ideally, if a wheel is perfectly circular and uniform, meaning its center of mass is exactly in the center of the wheel, and the axle for the wheel goes through the center of the wheel, then the wheel would be “balanced” because its center of mass and axis of rotation are in line (both at the center of the wheel). However, in the real world, these ideal cases are few and far between. Imbalances on a wheel, if they are drastic, can cause undesired vibrations.

Now, why does an imbalanced rotor tend to have large vibrations at certain speeds? The answer comes from the principles of resonance and resonant frequency. Every object has a resonant frequency at which the object experiences large vibrations or large-amplitude oscillations (resonance). Consider a small wine glass. If you drive the wine glass at its resonance frequency long enough (perhaps by directing sound waves at it), it will shatter due to the resulting resonance. Similarly, for an imbalanced rotor (such as a wheel), there are certain speeds of rotation (called “critical speeds”) that cause large-amplitude vibrations in the rotor (up-and-down and side-to-side). If one changes the speed of rotation away from this critical speed (either higher or lower), then the resonance will die out and the rotor will rotate more smoothly. Interestingly, at very high speeds, an imbalanced rotor will tend to balance itself out and rotate smoothly as if it was balanced to begin with.

When driving a car with imbalanced wheels at its critical speed, you might notice that the resonant vibrations feel uncomfortable to you, especially since you can feel the vibrations  from the steering wheel, which you are probably holding onto if you are moving on the road. Similarly, resonant vibrations are uncomfortable for your car as well (think of the wine glass analogy again). Prolonged periods of driving a car with an imbalanced wheel at its critical speed will subject the car to increased stress and cause it to wear out faster (and potentially break down as a whole due to failure under stress…just like people sometimes do). So, if you ever experience unusually large vibrations from your car when you are driving at particular speeds, you will know what the problem most likely is, and the strategic play would be to get the wheel imbalance fixed as soon as possible to avoid excessively damaging your car.