What does a string vibrating at a frequency of 196Hz look like visually?

[deadstar33]

I set up a simulation of a vibrating guitar string after it has been impulsed (plucked). It is in tension, and all of the parameters are set up so that it vibrates at a fundamental frequency of 196Hz, and the software has confirmed that this is indeed the fundamental frequency of the string at this particular tension I have set it to.

When I watch the simulation play out over the course of 1 second, the entire string is seen to vibrate at its fundamental mode as expected, and it also progresses through some of its natural harmonic mode shapes. However, the entire string only appears to move over and back (i.e. complete a full cycle of oscillation) only about 4 or 5 times over the course of the second. As I am not entirely certain what a string vibrating at 196Hz is supposed to look like, I'm not sure if the simulation is correct or not. The software says its fundamental frequency is 196Hz when it calculates it, so I expected to see the entire string move over and back 196 times. Am I correct in expecting to see this? I know that the string should be simultaneously vibrating in a pattern of different frequencies - the fundamental one as well as a combination of its natural harmonics.

Basically I'm not sure what I'm looking at and am trying to understand it, so if anyone is knowledgeable in this area I would really appreciate your input. Thanks.

 

[AlephZero]

In "real life" the string would look like a fuzzy blur. Your eyes can't follow the motion of anything vibrating as fast as that.

The image on your PC monitor probably only changes about 70 or 80 times a second. To see the details of something vibrating at 196 Hz it would need several images in each 1/196 of a second, so you would need maybe 2000 images a second - but there is no sense in doing that because your eyes can't follow the information at that speed.

The sensible thing to do would be slow the display down by a factor of 100 or even 1000, so you can see the details of what is going on.

EDIT: I just read your post again and maybe you are slowing it down already, but you are not creating enough separate images. About 2000 images per second of "real time" would be a reasonable starting point.

 

[deadstar33]

You were right, I had it slowed down but it was only running at 200 frames per second, giving the illusion that it was oscillating much less. I ramped the fps up to 4000, and now I can see each cycle playing out very clearly. Thanks for that.

While I'm on this topic, do you know if it is possible to calculate the damping coefficient between the string and the air? I've been trying for ages but I think I need a damping constant "c" for the calculation, and I don't know what that is or how to obtain its value...

 

[AlephZero]

Trying to calculate the damping coefficient directly from the physics is very hard. Instead of doing that, you can get the right order of magnitude by thinking about how much of the strain and kinetic energy in the string is converted into other forms (sound, heat, friction of the string over the guitar bridge as it vibrates, etc).

You can write the equation for simple harmonic motion without damping as
x'' + w^2 x = 0
where ' means d/dt and w is the frequency of vibration in radians/second.

In "physical units" the equation x'' + w^2 x = 0 is really
m x'' + k x = 0
where m is the mass of the string and k is its stiffness (which depends on the tension, of course), and
w^2 = k/m.

A solution to these equations is the sine wave
x = a sin wt
where a is the amplitude.
The energy in the string = strain energy + potential energy
= kx^2 /2 + m(x')^2 /2
= (1/2)k a^2 sin^2 wt + (1/2)m a^2 w^2 cos^2 wt
= (1/2)k a^2 sin^2 wt + (1/2)m (k/m) a^2 cos^2 wt
= k a^2.
In other words the energy is constant, which means the vibration continues for ever at constant amplitude.

The equation for damped vibration is of the form
mx'' + cx' + kx = 0
You know that mx'' and kx both have represent forces, so cx' must also be a force otherwise the equation would not make any sense.
So if we assume the damped vibration is approximately the same as the undamped vibration (which is OK so long as the amount of damping is small) we can find the amount of work done by the cx' force in one cycle of the vibration. This is the energy converted into sound, etc.

Work = integral of force x distance over one complete vibration cycle.
x = a sin wt
x' = a w cos wt

Work done = integral over one cycle of force d(distance)
= integral cx' dx
= integral cx' dx/dt dt
= integral c(x')^2 dt
= integral from 0 to 2pi/w c a^2 w^2 cos^2 wt dt
= c a^2 w pi

So the energy lost through damping in one cycle is approximately
c a^2 w pi.
Compare that with the total energy in the string = k a^2. Suppose a fraction b of the energy is lost in each cycle.
b = c a^2 w pi / k a^2
b = wc pi / k
b = sqrt(k/m) c pi / k
b = c pi / sqrt(km)

c = b sqrt (km) / pi

So, suppose you think your note of the guitar will last say 5 seconds before it dies away. 5 seconds is about 1000 cycle at 196 Hz. So the fraction of the energy lost in each cycle is about
1/1000, or b = 0.001 in the formula for c.

Actually, since the decay is exponential not linear, you will find the vibration takes more than 5 seconds to die out, but this will get you the right order of magnitude for c. You can then adjust it by trial and error to get the simulation working how you want it.

 

[deadstar33]

Thank you very much, that helps a lot. Very rigorous.

 

[AlephZero]

It's not really very rigorous, and there is a lot more to the subject of "damping" than what is in that post! From your OP I guess you wanted something where the "answer" was simple to use and the video of your simulation would look sensible, rather than a super-accurate model. If you use this math to generate an audio file instead of a video, it won't sound much like a note played on a real guitar.

If you want to see more theory, Google for the response of damped single-degree-of-freedom (SDOF) and many-degrees-of-freedom (MDOF) systems.