Archives For Engineering

Pops, Shocks, Impulses

January 8, 2014 — 1 Comment

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Take a balloon, blow it up, and quickly stick a needle through the rubber. What do you hear? You wouldn’t be surprised to hear a loud “pop” immediately after piercing the once-inflated balloon. Try it again with a few balloons of different radii. What do you hear now? Chances are that you will notice a slight change in frequency. Additionally, depending upon the room in which you pop the balloons, you may notice additional reverberation.

What is happening here? Why do different balloons sound slightly different? What is the reverberation, and is it useful?

First, why does a balloon make a loud popping sound? When a balloon pops, the rubber suddenly contracts. This leaves a discontinuity in the air pressure. Pressure outside the balloon is equal to the atmospheric pressure, but the balloon’s internal pressure is often a couple hundred Pascals higher than that of the surrounding air. Upon retraction of the rubber, this high pressure region meets the lower pressure of the atmosphere. This newly-formed pressure wave spreads outward from the center of the late balloon’s location as a weak shock wave. This abrupt change in pressure, as it spreads outward, acts like an impulse in the air, a point we will return to later. The balloon’s weak shock wave is similar to the strong shock wave from a jet plane, though the equations that govern the two differ.

As the peak in pressure propagates outward, something more fascinating is unveiled. Air is accelerated outward due to the sudden difference in pressure, and it will overshoot due to inertia. This leaves a region of low pressure behind the high pressure wave. Air will then accelerate inward in response and will once again overshoot, but this time it will do so in the opposite direction. The process continues, creating an oscillation in the air with a characteristic frequency. The frequency of this oscillation depends upon the radius of the former balloon. Thus, smaller balloons will have a more “shallow” sound, and larger balloons will sound “deeper.”

The question becomes far more interesting when considering that initial weak shock wave. As mentioned previously, this discontinuity acts like an impulse in the air. Impulses are powerful tools in that they contain all frequency information. If one wished to find the resonant frequency of a room, one could play sounds at various frequencies and find those which reflected most loudly off the walls of the room. However, this is a time-consuming process and is by all means impractical. An impulse, however, contains all frequencies. If an acoustical engineer were to supply an impulse at different locations in a room and place a microphone somewhere else, that engineer could calculate which frequencies are best reflected/selected by that room’s architecture. This could be done by firing a starter pistol or by clapping one’s hands (try it out). One could also pop a balloon. The balloon’s pop provides an impulse. The room (unless it is anechoic) will respond at particular frequencies. What this means is quite fascinating. The sound of the balloon’s pop is the sound of a recording studio, the sound of a theater, the sound of a living room, and the sound of a cafeteria.

The pop is the sound of the room itself.

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A Musician on Mars

December 11, 2013 — 1 Comment

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Welcome to Mars. As one of the first colonists on the fourth planet from the Sun, you endeavor to make it your new home. On Earth, you filled your time in numerous ways, but your real passion was music. Luckily, the Indian Space Research Organisation (ISRO) allowed you to bring your prized possession: a Steinway grand piano. Excited to play for the first time in months, you squeeze into your ISRO-issued space suit and wheel the piano onto the Martian surface. It’s noon near the equator. The temperature is around 25ºC (77ºF). You stretch out your arms, relax, and strike your first key. The sound is… quiet and out of tune. Assuming the piano needs to be retuned, you wheel it back into your pressurized vessel, take off your suit, and tune it yourself. Satisfied, you wheel the piano onto the surface again. The Martian surface is quiet, and you notice the colors of the sky are a lot redder than you had seen in NASA photographs. Again, you begin to play. It again sounds too quiet.

What is happening here? Why might a piano sound different when played on the Martian surface? This is a fairly involved question. Luckily, we are considering an instrument with taut strings rather than something that depends more upon atmospheric conditions than, say, a trombone or pipe organ. Furthermore, the equatorial temperature is Earth-like. Why, then, might a piano sound different on Mars?

When tuning and subsequently playing a piano, the frequency you perceive (or pitch) depends upon the tension, length, and mass of the strings within the piano. Since the temperature is about the same as before, and since you did not physically exchange the strings, these properties remain fairly constant. However, the fluid on the strings does play a role. Like any oscillator, the fluid in which it is immersed provides a load which will subsequently alter the frequency at which the oscillator resonates and by how much. On Mars, the atmosphere is more rarified, with a mean pressure of 600 Pa at the surface. Compare this with a pressure of over 100,000 Pa at sea level on Earth. This reduced loading by air results in a bias to slightly higher frequencies (or a higher pitch). If you retuned the piano in a pressurized cabin and then played the newly tuned piano on the Martian surface once again, it would still sound out of tune. A simple solution is to retune the piano while on the surface.

However, this is not the only problem with playing music on the Martian surface. Remember that Mars has a lower-pressure atmosphere. Sound, as you may recall, propagates as an oscillation of pressure in some medium (like air). If the mean pressure is lower, this presumably changes the ability of sound to propagate over longer distances. Without going into too many details here, what happens is that sound will not propagate very far on Mars, and there is an effect such that high frequencies are heavily attenuated. Before, the pitch was shifted slightly higher. Here, on the other hand, higher frequencies will sound softer than lower frequencies, and all frequencies will sound quieter. This means that not only does the piano sound out of tune, but it also sounds muted. The question of sound propagation is so interesting that an acoustics researcher simulated sound on Earth, Mars, and Titan. She found that a scream which may travel over one kilometer on Earth would only carry 14 meters on Mars!

Your out-of-tune, muted piano, probably wouldn’t be audible to a nearby audience on the Martian surface.

Plug Me In

October 8, 2013 — Leave a comment

I’m going to have a bit more fun with this blog post. For this thought experiment, I’d like you to suspend your disbelief. Imagine, for a moment, that someone offered you the chance to “plug” your body into a standard outlet and let yourself “charge.” All of your energy would be gathered from this charging process. You would eat nothing. How long must you remain connected to the outlet? How much will it cost?

Where do we start? There are a few ways to approach this, but I’ll start with the basal metabolic rate for an average adult male. For a 70 kg male, this is typically around 1,600-1,700 Calories (kilocalories). If you would like to do more than just sit against a wall, you will need a bit more energy. Let’s round that up to 2,000 Calories. Converting this to units with which we can work, this comes to 8.36 megajoules (MJ). Like most thought experiments, it is easier to work in orders of magnitude, so we will round this up to 10 MJ.

We now know how much energy we need, but how long will it take to draw this energy from an outlet? Every outlet has a maximum power draw, but very few appliances, if any, reach this maximum value. We denote the amount of power drawn in joules per second as Watts (W). On average, microwaves draw 1,450 W, vacuum cleaners 630 W, computers 240 W (though, as I type this, I am drawing <100 W), and alarm clocks 2 W. In other words, it’s variable. If we were to charge ourselves like a microwave oven, it would take almost 2 hours. However, if we used a computer charger (100 W), it would take 28 hours! A laptop computer charger would thus not suffice, since we would not acquire our necessary daily energy within a given day. All of this energy would be expelled as heat, and you would be a blob of meat plugged into a wall outlet. That’s not a fantastic way to live.

In case you are wondering, architectural engineers model heat production from humans as if they were 100 W light bulbs. This is eerily similar to our 100 W laptop charger that provides just enough energy to get us through a single day!

If you tried all of the above with one of Tesla’s new 10 kW chargers, you’d be ready for your day in only a few minutes!

What about the cost? Two apples provide approximately 200 Calories of energy (note that the energy yield from eating is not 100%, so you will actually receive less than 200 Calories from an apple). The cost of the apple varies based upon season, region, type, and quality of the fruit. Let’s say the two apples cost you $1.00 for ease of comparison. You spend $1.00 for 200 Calories of fresh, delicious apple. How does this compare to energy cost from your wall outlet? In the United States, the average is $0.12 per kWh. The energy from those apples, then, would cost you less than three cents. Over the course of a year, you would spend less than $200 to keep yourself more than fully charged! Imagine spending that much on food in a given year.

Do not try this at home, or while in the Navy.

Here’s a conundrum for you: Using only technology available hundreds of years ago, how could you determine the speed at which light travels? We know now that light travels at 299,792,458 m/s, or, to put it simply, “very, very fast.” In fact, we are so sure of this value that we use it to define the meter, where one meter is equal to the distance that light travels in 1/299,792,458 of a second. Today, we have access to technology which allows us to calculate this value. Time-of-flight devices pulse bright flashes of light which are reflected off a mirror, and the difference in time (down to nanoseconds) combined with the distance from the source/detector and the mirror provides an accurate measurement of the speed of light. Additionally, one can take advantage of cavity resonators or interferometers to obtain the same value. However, these devices did not always exist, yet estimates for the speed of light predate their existence. How was this accomplished?

In a first account of the discussion on light propagation, Aristotle incorrectly disagreed with Empedocles, who claimed that light took a finite amount of time to reach Earth. Descartes, too, claimed that light traveled instantaneously. Galileo, in Two New Sciences, made the observation that light appears to travel instantaneously, but that the only observation is that light must travel much faster than sound:

Everyday experience shows that the propagation of light is instantaneous; for when we see a piece of artillery fired, at great distance, the flash reaches our eyes without lapse of time; but the sound reaches the ear only after a noticeable interval.

To determine the speed of light, Galileo devised a time-of-flight experiment similar to the one described above, where two individuals with lanterns would stand at a distance, uncover and recover them upon seeing a flash from the opposing partner, and calculate times between flashes. By starting very close to account for reaction times and eventually moving very far away, one could see if there is a noticeable change in latency. However, this experiment is challenging, to say the least. Is there a simpler method?

Enter Danish astronomer Ole Roemer. Known in his time for accuracy in measurement, arguments over the Gregorian calendar, and firing all the police in Copenhagen, he is best known for his measurement of the speed of light in the 17th century.

While at the Paris Observatory, Roemer carefully studied the orbit of Io, one of Jupiter’s moons. Io orbits Jupiter every 42 and a half hours, a steady rate. This discovery was made by Galileo in 1610 and well-characterized over the following years. During this time, Io is eclipsed by Jupiter, where it disappears for a time and then reemerges sometime later. However, Roemer noticed that, unlike the steady state of Io’s orbit, the times of disappearance and reemergence did change. In fact, Roemer predicted that an eclipse in November 1679 would be 10 minutes behind schedule. When he was proved right, the Royal Observatory remained flabbergasted. Why was this the case?

The figure above, from Roemer’s notes, highlights Earth’s orbit (HGFEKL) around the sun (A). Io’s orbit eclipses (DC) are shown, defined by Jupiter’s (B) shadow. For a period of time, at point H, one cannot observe all eclipses of Io, since Jupiter blocks the path of light. However, when Earth is at positions L or K, one can observe the disappearances of Io, while at positions G and F, one can observe the reemergences of Io. Even if you didn’t follow any of that, note simply that while Io’s orbit does not change, the Earth’s position relative to Jupiter/Io does change as it orbits the Sun. One observing Io’s eclipse at point L or G is closer to Jupiter than one observing an eclipse when the Earth is at point K or F. If light does not travel instantaneously, observations at points K and F will lag, because light takes a bit longer to reach Earth from Io.

In order to calculate the speed of light from this observation, Roemer needed information from his colleagues on the distances from the Earth to the Sun. Additionally, there are other complications. Nonetheless, using the measured distance from the Earth to the Sun at the time (taking advantage of parallax), Roemer announced that the speed of light was approximately 220,000 km/s. While more than 25% lower than the actual speed of light, it remains astounding that one could estimate this speed using nothing but a telescope, a moon, and a notebook.

Giovanni Cassini, a contemporary of Roemer, was not convinced at first. However, Isaac Newton noted the following in his Principia, from Roemer’s observations:

“For it is now certain from the phenomena of Jupiter’s satellites, confirmed by the observations of different astronomers, that light is propagated in succession and requires about seven or eight minutes to travel from the sun to the earth.” 

In other words, philosophers now began to accept that light travels in a finite amount of time.

Over the course of many years, others continued to estimate the speed of light using creative methods. James Bradley, in 1728, noticed that the positions of stars changed during rainfall, using these observations to estimate the speed of light with great accuracy (Bradley: 185,000 miles/second; Speed of Light: 186,282 miles/second). In 1850 in France, Fizeau and Foccault designed a time-of-flight apparatus like the one described in the opening paragraph. As opposed to using modern technology, the apparatus uses a rotating wheel to simulate blips of light. With a wheel of one hundred teeth moving at one hundred rotations per second, the speed of light could be calculated to within the accuracy of Bradley’s observations. Albert Michelson, in the 1870s, repeated the measurements on a larger scale, again with a series of mirrors.

What can be gleaned from this story is a powerful lesson. At times, the simplest observations can result in the most compelling findings. What it required in this case was careful note-taking and a bit of intellect. Even without those, simple observation cannot be understated.

Keeping it Random

January 7, 2013 — Leave a comment

When iTunes “shuffle” was introduced, Apple received many complaints. It turns out that a number of songs were played many times, and customers felt that the randomness of this random shuffle algorithm was not truly random. Apple changed the algorithm, and it works a bit better now. However, their change actually made the process non-random. The previous iteration of the software was random. Why, then, did the complaints arise?

If you take a carton of toothpicks and throw them across the room in a truly random manner, you will notice that the toothpicks will start to form clusters. This “clumping” occurs due to the nature of a Poisson point process, or a Cox family of point processes. Simply put, the process tends to create clusters around certain locations or values when it is truly random. The same also occurred in World War II. The Germans were randomly bombing Britain. However, the randomness led to the same type of clustering one would see in iTunes. Certain targets were bombed more often than others. This led the British to think that the Germans had some strategy to their bombing when, in fact, the process was purely random. We tend to think that a random process would be evenly distributed, and when the reality defies our logic, we no longer see the randomness in the random process. Apple decided to change their algorithm to a less random but more evenly distributed one, and customers remained happy.

I can discuss different types of randomness fairly extensively, but I would rather touch upon two different types of random number generation. These are pseudo-random number generators and true random number generators. Pseudo-random number generators use mathematical formulae or tables to pull numbers that appear random. This process is efficient, and it is a deterministic, as opposed to a stochastic, one. The problem is that these generators are periodic and will tend to cycle through the same set of pseudo-random numbers. While they may be excellent for pulling random numbers on small scales, they fall prey to significant problems in large-scale simulations. The lack of true randomness creates artifacts in data and confounds proper analysis.

True random number generators, on the other hand, use real data. Typically, data from physical observations, such as weather patterns or radioactive decay, are extracted and used to generate random values. The lavarand generator, for example, used images of lava lamps to generate random numbers. These true random number generators are nondeterministic and do not suffer from the periodicity of pseudo-random number generators.

This distinction is important in the simulation of data. How can one best generate random numbers? If an internal clock is used to generate random numbers, but you are iterating through some code thousands of times, a periodicity dependent upon the computation time may result and generate artifacts. The use of atmospheric noise could overcome this, though pulling the data takes time and could slow down computation.

The world around us is filled with processes both random and nonrandom. It is a challenge to generate artificial random processes, and it is surprising that truly random processes often appear nonrandom to human observers.

A couple weeks ago, Felix Baumgartner set the record for the longest free fall, previously held by Captain Joseph Kittinger. To be more specific, Baumgartner dove from 39 kilometers in 2012, and Kittinger dove from 31 kilometers in 1960. However, Baumgartner traveled faster, with his total dive taking 17 seconds less than Kittinger’s. This was the true feat, as he reached 1,342 kilometers per hour, thus breaking the sound barrier.

At 39 kilometers, how high up was Baumgartner? This is 8% short of a full marathon, the distance of 4.4 Mount Everests stacked atop one another, and 3.6 times the greatest depth of the ocean. At this height, the temperature is only -25.6 C, the pressure is only 1/3 of that on the ground, and the effect of gravity was still 98-99% of what it would be at sea level. His maximum speed of 1.1 Mach (the speed of sound in dry air at 15 C and 1 atm) was just 13% short of the maximum speed of the X-1 rocket plane. In other words, he was moving very fast from an extreme height.

If you want to consider how this might affect a human, you must consider not speed, but acceleration. With the Stratos jump, it took Baumgartner 42 seconds to reach his terminal velocity. This is pretty quick. How did he do it? When in free-fall, we can consider two forces, drag and gravity. I noted above that the effects of gravity were reduced, but not by much, at this elevation (to 9.7 meters per second-squared, to be specific). The force of drag acting on a body is dependent upon its velocity. So, as you fall and gain speed, the effects of drag become greater. Eventually, drag force becomes great enough such that you cannot accelerate any further from gravity, and you reach terminal velocity. This is usually 25 m/s for most objects. However, Baumgartner was in the stratosphere. The air pressure, as I mentioned above, was only 0.33 that of what is at sea level. With such low air density, the effects of drag are reduced. The force of acceleration, relatively unchanged, provides a strong downward force (toward the Earth). This leads to a very high terminal velocity, one that can break the sound barrier. (I should note that the speed of sound is reduced at high elevation because sound propagation is dependent upon the density of the medium through which the wave travels.)

What does this have to do with acceleration, then? We know his speed was significant, and we know he did it quickly. According to the argument above, the only real force leading to acceleration is gravity, and we have an upper limit of 9.7-9.8 m/s^2. The numbers reported agree with this, with an average acceleration of about 8.8 m/s^2. Is this fast enough to hurt a human? Again, what matters is accelerationnot speed. This discussion started in WWI, where pilots reported vision problems with high-acceleration maneuvers. Today, we see it ranging from the design of rockets by NASA to safety reports in four-door sedans. If we look back at other human endeavors, we see a story of high acceleration. The now-retired shuttle missions accelerated astronauts to three times the force of gravity. The Apollo missions entered the atmosphere at six times the force of gravity upon their return home. However, the highest reported cases were with the Daisy Decelerator in the 1970s. Major Beeding was placed in this capsule, and he was decelerated at 83-times the force of gravity for approximately 0.04 seconds. He survived, emerging with a short period of shock and a bruised back. While I wouldn’t claim that most humans could survive such an extreme scenario, this demonstrates the importance of acceleration over total speed. With Baumgartner traveling at <1g (less than one times the force of gravity), this provided little danger.

Whether or not this provided a danger to the diver, it was an exciting watch. Whether or not it advanced our knowledge of the stratosphere, it gave me a fun topic for this blog post.