Saturday, May 18, 2013

So quiet, a pin drop sounds like a plane taking off...

This has been a hell of a week. Some folks came in from out of town, most notably Dr. Pat Walter from TCU, who came in to give a lecture on how to take better shock and vibration measurements. Pat draws a pretty good crowd, about 30-40 people from different industries who are all interested in dynamic measurements. Among these was an old friend of mine, Andy, and some of his colleagues from Los Alamos National Lab. I was between hanging out with them, and dealing with some other boring, non-work related stuff all week, hence no posts. Sorry folks, that will happen from time to time...


Anyway, I'm going to steer a bit off of the lesson plan just a bit (And actually this will fit in nicely with sinusoidal amplitude, which I'll be talking about next) and talk about an application. Yes, I know that it's about time, but the fundamentals must be learned before we can actually understand the applications and all of the cool things in dynamics.

Orfield Labs in Minneapolis, MN had Eckel Noise Control Technologies design and build what Guinness World Record's calls the quietest place on Earth. It's an anechoic chamber (an meaning not, and echo meaning a sound reflection) within two more nested rooms. The nested rooms isolates the anechoic chamber from the outside world to such an extent that the background noise level of the room with nothing in it except the measurement equipment is -9.4 dBA (Decibels, A-weighted). The room is pictured here:


I know a lot of folks have heard the term dB thrown around a lot, so let's talk about what that actually means:


The decibel is a logarithmic unit that indicates the ratio of a physical quantity (usually power or intensity) relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities.(Wikipedia)

So, simply put, a dB is an indication that what's being measured is in relation (a ratio) to something else. In power quantities (watts and volts^2), an increase in +10 db means a times 10 increase in the actual number. However when dealing with the root of power quantities like pascals, volts, m/s^2... (pretty much everything that isn't watts or volts^2), an increase of 20 is needed for a times 10 increase in the quantity, usually an RMS (Root, Mean, Square) level.

Since a dB is a ratio, or the quotient of two numbers, and the quantity we are measuring is the dividend, what's the divisor, or the reference of the ratio? In each case it can be different, but for sound pressure level, the reference is the lower threshold of human hearing, which is 20uPa. This means that a person with absolutely perfect hearing cannot distinguish an acoustic pressure variation smaller than 20uPa. This, then, is 0 dB. 

The threshold of human hearing isn't a very good means of comparing sound on a day to day basis. Human breathing creates about a 10-20 dB sound pressure level. Normal conversations have a sound pressure level of between 40 and 60 dB. Most automobiles are between 60 and 80 dB. Hearing damage starts to occur at around 85 dB. 

What this means for the anechoic room at Orfield Labs is that when no other stimulus is occurring within the room, the sound level is more quiet than what a human can perceive. In fact, the background noise level in this room, at -9.4 dBA means that the sound pressure is about 1/3 of the pressure we could perceive. In a room like this, a person can hear a lot of things about themselves that they couldn't normally hear. Your heart beating, the gurgling of your stomach, creaks in your joints, and even the blood rushing through your own head. It can be said that when you enter this chamber, you become the sound...

So... Why build a room like this? What's the purpose? I'll let you know soon!

-CMW

Friday, May 10, 2013

What goes into a sign?

Sine that is... a sinusoid... That is where we left off, correct? (Looks back at Up, Down, Up, Down...)

Let's go back a little bit first. What are sine (and cosine)? They represent a function where the output is the ratio of one of the two legs of a right triangle over the hypotenuse. You say, "This sounds a LOT like trigonometry," Well that's where a LOT of our math in engineering comes from. Sine is the ratio of the leg opposite the measured angle and the hypotenuse, which, when the angle is zero, the length of that side is zero. In this instance you really don't have a triangle, you have a line segment the length of the hypotenuse, but that's not a big deal. Maybe a picture is needed for this.
This is called the Unit Circle. It's made by changing the angle that the hypotenuse of a right triangle make with the x-axis, while keeping the radius one (hence unit). Looking at sine (the vertical component of the triangle we see that as we increase the angle, the ratio gets bigger until the angle measured reaches a right angle, 90 degrees, or pi/2 radian (we'll probably stick to radians a lot, since the math later will be much easier that way). At this point, sine begins to shrink again until and angle of pi, where it becomes zero. 

Now, on the lower half of the circle, we see sine grow negative, to a maximum of 1 at 3pi/2, and then shrink back to zero at 2pi, or 0 radians (same thing). If we were to plot each of these points as we went around the unit circle, we would get a plot of what we call a sinusoid. Wikipedia has a great graphic for this:

Now because the function's parameter, theta, is locked in to radians, we can adjust the scale of sine across other variables. This parameter is called the phase of the function. Commonly we use space and time variables (x,y,z, and t) to scale sinusoids across those domains. This is shown below:
The higher the frequency, the more cycles per duration of the domain the sinusoid goes through. In terms of acoustics, the higher the frequency, the higher the pitch of the sound we're hearing. For those music types, Middle C is about 264Hz. We'll talk about frequency more later. Because the shape of the sinusoid is being stretched over the domain we're working over (x,y,z, or t) we want to be able to measure that as well. If we measure the distance between consecutive peaks or troughs, we get the wavelength, often stated with the variable lambda:
Wavelength can bring on many discussions of speed of a wave in a medium (Speed of sound or Speed of Light), as well as a whole other realm of mathematics. Needless to say, we're going to hold off on that. Speed of sound will come soon enough.

Shifting a sinusoid can be done by adding a number to the phase, this then called a phase shift. Here's an example of that:
This will become very important later when we want to compare to signals, and when we want to talk about physical phenomena like resonance and electrical phenomenon like filtering.

I'm going to save talking about the amplitude of a sinusoid until next time, since there are a ton of ways to describe it, and getting the all straight is something that I know many people have a hard time with.

Sorry about the delay in getting this out, real life caught up with me this week, and I was super busy at work, which made me crazy tired when I got home, and therefore crashed early, when I should have been writing this...

Anyway, if this weekend isn't too nice, I'll try to get back to amplitude on Sunday! Cheers!

-CMW

Thanks to Wikipedia for use of all of these images. Without you, I would have had to generate them in MATLab. I believe that all of them are linked properly, so you can check out the various Wikipedia sites where they reside.


Thursday, May 2, 2013

Up, Down, Up, Down...

This is

y=sin(x), courtesy of Google.com. This is actually one of Google's cooler features, the ability to do simple plots from the search box. I highly recommend trying it out! 

No worries, touting the much awesomeness of Google is not why I'm here tonight.

The title describes it pretty well I think. There are many things that go up and down in our world: the light switch in your office, the bumps on the road, the hands on the clock (well if you just look at the motion of the tip of the hand, there's up/down there), and the function shown above. You don't just have to limit it to up/down, you can also say forward/back (like a playground swing), or left/right (like a pendulum and the tips of the hands of the clock, again).

If you think about sound, and one of the fundamental sources of sound, a speaker, you come to realize that the motion of the speaker is forward/back (depending on how you view it). The diaphragm of the speaker moves back and forth depending on the signal coming to it from the amplifier (Electrodynamics of a voice coil is another blog post...way down the road), and this in turn pushes the air molecules back and forth, creating a sound wave that propagates. This sound wave can reach someone's ear, where it can be heard (Might get to hearing at some point in the future, it's a bit of an interest to me).

So now we know what the motion looks like, how do we write that down? Typically words are great at describing fixed or static things, but nobody wants to type the position of a speaker diaphragm as it is moving. First, you would be writing until the object stopped moving. Second, no one can type fast enough while measuring the speaker diaphragm position in order to do it accurately. Third, even if you get the position correct, when was it correct? 

Luckily, us folks with a math background have functions! We can describe a wide variety of shapes, complex geometries, solids, curves, and surfaces using mathematical functions. The cool part, and the part I want to talk about on here a lot, is how to describe motion. This is, after all, the most fun with a piece of chalk! (If you know what that means, please feel free to comment! Guesses are welcome too!) (Nick M, you're not allowed to guess, neither are you Peter R, if you read this.)

I plotted sin(x), which is the vertical position of y, when y is the sin(x). This makes it valid for any two dimensional set of labeled axes. (Note I said labeled, this is a pet peeve of mine...) However this can be made just as valid for y=sin(.08*pi*t), where t is time, let's assume in seconds. (Yes, I know I added a parameter, I'll get there soon!) Let's also assume that y is something we can measure, like the position of the speaker diaphragm, in inches. This position can be measured with sometime like a laser used by a contractor to measure the rooms in a house, except really accurate, and it can measure it over a constantly repeated time span (Once per second... Ugh, Sampling, the topics keep piling up) This would look a little like this:



Here's the nifty part, we can use that measured function to replicate it, and in turn get an image of the original y=sin(.08*pi*t). In fact, we do it all the time! We measure tons and tons of complex things, record them, store them, transmit them, reconstitute them, and play them back (probably though a speaker). However in order understand the content of the things we measure, we have to compare them to some reference. Because the general motion of a lot of stuff is back/forth, left/right, up/down, a sinusoid is a great way to do this.

So what was with that extra bit I put into the sin() function above? I'll save that for next time...

In case you guys haven't noticed, I'm trying to do this Tuesdays and Thursdays, and I might sneak in a Sunday here or there, depending on if I'm home and how much other crap I have to do. I see people taking a peek, but still no comments (but a couple +1's on Google+). I know I'm pretty light on these topics (for some people...). If something gets too complex too quick, drop me a message on here and I can try to explain things better.

-CMW