How do macroscopic brownian ratchets extract energy from random vibrations?

[parsec]

I have been noticing a few examples of systems that can extract energy or do useful work from seemingly random vibration, and I can't figure out why this is.

If you take a guitar and smack its body, the strings will ring out to some extent. From a spectral point of view, the delta function excitation of the body should contain a wide band of frequencies that may feed a small amount of energy into taught strings, which are a high Q resonant system.

I think I am okay with this, because the excitation is unipolar. From a time domain perspective, it is not so dissimilar to plucking a string, which is a delta function displacement of the string, which then causes it to ring out at its resonant frequency. In effect, the short length of the excitation pluck/smack might guarantee that the string is in phase with its forcing function.

I did an experiment where I took a shaker and attached it to a guitar body. A shaker is essentially an electromagnet that is configured to displace a large mass back and forth. I applied amplified white noise to the shaker, and this induced a weak excitation of the strings. Low pass filtration of the white noise caused the strings to ring more strongly.

This feels like a kind of macroscopic brownian ratchet. The excitation from the shaker is bipolar, so one would think that for every bit of vibration that can force a string to displace and oscillate, there should be some vibration that counteracts this oscillation... yet somehow the strings can collect a net amount of energy from this random excitation. There is some kind of mechanical diode at play.

Why is this the case? Is there something in the way that the vibration of the guitar body couples with the strings that causes the strings to more efficiently pick up energy during certain parts of their oscillation cycle?

I am wondering whether the same thing might occur if I were to apply white noise to an electromagnetic driver to directly move the strings rather than vibration to the guitar's body.

Incidentally, I noticed an interesting mechanical analog of this system. It seems to exploit asymmetries in excitation to induce unipolar motion (rotation in this case).

http://en.wikipedia.org/wiki/Gee-haw_whammy_diddle

 

[Greg Bernhardt]

I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?

 

[Dale]

I did an experiment where I took a shaker and attached it to a guitar body. A shaker is essentially an electromagnet that is configured to displace a large mass back and forth. I applied amplified white noise to the shaker, and this induced a weak excitation of the strings. Low pass filtration of the white noise caused the strings to ring more strongly.

This feels like a kind of macroscopic brownian ratchet. The excitation from the shaker is bipolar, so one would think that for every bit of vibration that can force a string to displace and oscillate, there should be some vibration that counteracts this oscillation... yet somehow the strings can collect a net amount of energy from this random excitation. There is some kind of mechanical diode at play.

Why is this the case? Is there something in the way that the vibration of the guitar body couples with the strings that causes the strings to more efficiently pick up energy during certain parts of their oscillation cycle?

From your description this seems to be a manifestation of the common phenomenon called resonance. A system which exhibits resonance has a few frequencies where energy transfer is very efficient, and energy transfer is inefficient at other frequencies.

If an input signal has a lot of frequencies, like a delta function or white noise, then the system will efficiently receive only the energy at the resonant frequencies.

 

[parsec]

I understand that from a frequency domain perspective, a resonant system should be able to harvest energy from a broadband excitation such as white noise... but from a time domain perspective, I don't understand how this works.

For every bit of vibration that can push a string one way, there should be a net amount of vibration capable of pushing it the other way as well. If simply being a resonant system could allow systems to harvest energy from completely random bipolar excitation, shouldn't strings start vibrating due to the thermal motion of air molecules as well?

I noticed another example of this kind of phenomenon. In our lab all of our high voltage capacitors must be shorted when kept in storage. Failure to do so can lead them to charge up over time. As far as I am aware, there are no temperature gradients in the room, so it feels like another successful variant of "maxwell's demon" that lies somewhere between microscopic and macroscopic.

 

[Dale]

The time domain perspective is pretty much the same as the frequency perspective. You just replace "frequency" with "period". Resonance is where a system has efficient transfer of energy from sinusoids of a specific period and inefficient transfer for other periods.

I am sure that the strings do vibrate from thermal motion of the air. But resonance doesn't create energy or even move energy from one frequency to another. It simply allows efficient transfer at certain frequencies. If there is very little energy at the specific frequency, like audio frequencies for thermal motion, then the resonant system will efficiently receive very little energy.

 

[AlephZero]

This feels like a kind of macroscopic brownian ratchet. The excitation from the shaker is bipolar, so one would think that for every bit of vibration that can force a string to displace and oscillate, there should be some vibration that counteracts this oscillation... yet somehow the strings can collect a net amount of energy from this random excitation. There is some kind of mechanical diode at play.

I don't think so. The main feature here is that the impulse response of the strings is very long. A random excitation of a mode in the time domain will tend to cause a random walk that builds up amplitude over time, without any nonlinear effects.

In fact the difference in impulse response time between the strings and the body of the instrument is used to get an efficient algorithm for physical modeling simulation of musical instruments. Instead of plucking the modeled string and coupling the long decay time to the response of the instrument, you filter the "pluck" through the impulse response of the instrument without any strings (which can be measured rather than calculated) and feed that into a relatively simple model of the vibrating string. In other words, if you hit the body of the guitar in a clever enough way for a short period of time (a fraction of a second), you would get a long sustained note from the string just as if you plucked it in the conventional way.

 

[parsec]

I am sure that the strings do vibrate from thermal motion of the air.

Yes, but if they can collect a net amount of energy in a resonant mode then the system violates the second law.

If there is very little energy at the specific frequency, like audio frequencies for thermal motion, then the resonant system will efficiently receive very little energy

My understanding is, if the system is high Q resonant, then that little energy fed into the system can result in large amplitude oscillations, only mediated by the system's damping (energy loss rate). When the energy loss rate comes into equilibrium with the rate at which the excitation is feeding energy into the system, the amplitude saturates. If your scenario regarding thermal excitation of the strings is true, then by reducing frictional loss from the system, the string should build up a non-zero vibration at its resonant frequency, in violation of the second law.

 

[parsec]

I don't think so. The main feature here is that the impulse response of the strings is very long. A random excitation of a mode in the time domain will tend to cause a random walk that builds up amplitude over time, without any nonlinear effects.

In fact the difference in impulse response time between the strings and the body of the instrument is used to get an efficient algorithm for physical modeling simulation of musical instruments. Instead of plucking the modeled string and coupling the long decay time to the response of the instrument, you filter the "pluck" through the impulse response of the instrument without any strings (which can be measured rather than calculated) and feed that into a relatively simple model of the vibrating string. In other words, if you hit the body of the guitar in a clever enough way for a short period of time (a fraction of a second), you would get a long sustained note from the string just as if you plucked it in the conventional way.

This is an interesting way of looking at it, and explains why low pass filtration of the white noise caused the strings to ring out more. When the shaker's excitation resembled a low rumbling vibration, it was most effective in exciting the strings.

The part I don't get is, when applying broadband white noise over a long period of time, the strings still do ring out a little bit. It's as if the strings start vibrating and then that vibration modifies the way they can collect energy from the white noise so that there is a net positive effect.

 

[Dale]

How would that violate the second law?

The same thing happens for gray bodies (I.e. not blackbodies). The second law doesn't forbid resonance.

 

[parsec]

The second law forbids a system from acquiring a net positive amount of energy from random thermal fluctuations just by virtue of it being resonant. The reasoning is that for every mechanism that can add energy to the system, there should statistically be an opposite mechanism to remove that energy from the system, as is the case with the brownian ratchet or any other Maxwell's demon type hypotheticals.

 

[Dale]

I don't think that is correct. The second law forbids a net energy transfer from a cold in object to a hot object at any frequency, but there is no reason that the transfer must be equally efficient at all frequencies.

I think you are simply misunderstanding the second law.

 

[parsec]

I don't think I am.

The second law also forbids a system from spontaneously acquiring a net amount of energy from random thermal fluctuations, regardless of its resonant properties.

The simple machine, consisting of a tiny paddle wheel and a ratchet, appears to be an example of a Maxwell's demon, able to extract useful work from random fluctuations (heat) in a system at thermal equilibrium in violation of the second law of thermodynamics.

http://en.wikipedia.org/wiki/Brownian_ratchet#Why_it_fails

 

[Dale]

The second law also forbids a system from spontaneously acquiring a net amount of energy from random thermal fluctuations, regardless of its resonant properties.

No it doesn't. A system can indeed spontaneously acquire a net amount of energy from random thermal fluctuations as long as the system is colder than the thermal bath. Resonance doesn't change that at all.

 

[parsec]

No it doesn't. A system can indeed spontaneously acquire a net amount of energy from random thermal fluctuations as long as the system is colder than the thermal bath. Resonance doesn't change that at all.

Obviously we are discussing systems that are at the same temperature.

 

[Dale]

Then energy will transfer efficiently at the resonant frequency in both directions for no net transfer. An efficient absorber is also an efficient emitter. (It certainly wasn't obvious to me that you were discussing thermal equilibrium)

Resonance changes the efficiency of energy transfer, not the direction. If it is efficient in then it is also efficient out.

 

[parsec]

So in conclusion, a string cannot start to spontaneously oscillate at its natural frequency due to thermal motion of air molecules around it, given the system is at the same temperature as the air.

You seemed to disagree with this sentiment initially.

 

[Dale]

If a string is not oscillating at its resonant frequency then it must be cold. If it is cold then it can certainly absorb thermal energy at the resonant frequency and begin oscillating.

If it is already warm then it is already oscillating at the resonant frequency as well as any other modes that it has. It will lose as much energy as it gains and continue to oscillate at that same amplitude.

I have not changed my position, it was not clear to me that you were talking about a system in thermal equilibrium.

 

[parsec]

I don't understand why you are introducing temperature at all. I never discussed temperature. It seemed pretty obvious that the system is in thermal equilibrium. If I were to talk about a system in a room and never use words like "hot" or "cold" or "temperature", wouldn't you assume the system to be in thermal equilibrium with the room?

 

[Dale]

A system which is warm will have oscillations in all of its modes. That is the equipartition principle.