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Analysis of Entrainment & Decoupling between a Thermoacoustic Oscillator & a Driving Force
Dakota Hall & Emmanuel Sanchez
Abstract
A thermoacoustic oscillator was observed to synchronize with an external sound source (i.e. speaker.) The coupling and decoupling rates were investigated by looking at the phase relationship between the thermoacoustic oscillator and the speaker. For the conditions with the right frequency differences and coupling strength, they were observed to couple extremely quickly if not instantly, however, decoupling was observed along the fringes of the region satisfying the conditions, especially when the speaker’s frequency was below the thermoacoustic oscillator’s resonant frequency. The range in which the systems coupled agreed with the Arnold’s tongue model. The temperature dependence of the shape of the Arnold’s tongue was also examined. This is an extension of a previous MXP project in which the phase relationship between the thermoacoustic oscillator and the speaker was measured.
Introduction
Synchronization is a widely known phenomenon whose impacts are visible in many different areas of study. For example, the chirping of crickets and the blinking fireflies synchronize with each other, creating biological coupling. Physical coupling causes pendulums of two grandfather clocks to synchronize when they are placed next to each other. [1] As such, synchronization of frequencies occurs for a system of a thermoacoustic oscillator with an external sound source. Thermoacoustic oscillators can be used to explain the underlying principles of a thermoacoustic engine, an engine studied heavily in the past three decades due to its high efficiency. [1, 2] The synchronization of thermoacoustic engines has been used to eliminate high-amplitude vibrations of pipes. [3]A thermoacoustic oscillator is a tube with porous ceramic stack, dividing the tube into two chambers. A heat gradient applied to the ceramic stack creates an airflow, generating a white noise that resonates in the tube with a fundamental frequency. If there is an external sound source that is close to the same frequency as the thermoacoustic oscillators, the frequencies can synchronize with each other. We will measure the amount of time it takes for the thermoacoustic oscillator to desynchronize with an external sound source of varying frequency for different intensities.In what follows, the theory of thermoacoustic oscillators and how it couples with an external sound source will be described. This will lead into our experimental setup, which will then be followed by our data and the analysis. A summarized overview of the project will be supplied at the end.
Theory
Arnold's Tongue
A standing wave can be reproduced inside the cavity of a cylindrical glass tube with one closed end, like the acoustic oscillator. The standing wave is generated by a heat gradient inside the tube that spans even integer lengths of the tube. This phenomena results in a natural resonant frequency in air produced by the speed of sound and geometry of the tube [5]. The fundamental frequency, or the first harmonic frequency, pertaining to the thermoacoustic oscillator is modeled by the relationship.
Where c is the speed of sound in air, and the length of the tube is L. The factor of ¼ is because the nodes of the wave occur after distances of 2L, which is only half of the full wavelength. This standing wave inside the thermoacoustic oscillator is illustrated in Figure 1
Figure 1. Illustration of fundamental standing wave produced inside thermoacoustic oscillator by a heat gradient. The microphone will be inserted through the middle of the stopper, keeping a tight seal.
The thermoacoustic oscillator will generate this fundamental frequency when sufficiently energized by the heat gradient of the ceramic stack. The heat gradient causes differences in pressure on each side of the ceramic stack, so the tube oscillates as air travels from the hotter, high pressure area to the cooler, low pressure area and creates white noise. Only the fundamental frequency corresponding with the geometry of the tube will be amplified and propagate as the resonant frequency.
An interesting question is how to describe the synchronization of the oscillator with the external frequency. The oscillator maintains its own resonant frequency but it can be influenced/driven to vibrate at a different frequency depending two conditions. These two conditions will be expressed through the concept of an Arnold’s tongue. Arnold’s tongues show the three states of the coupling strength vs frequency plane. The inner state (I) is complete synchronization, the outside (III) of the tongue is no coupling and the yellow colored fringes (II) describe the transition between coupling and decoupling. On the x axis is the difference in frequency (∆f), and the y axis is the coupling strength coefficient, (b). They encompass the frequency difference and coupling strength combination that results in coupling between the thermoacoustic oscillator and the external noise source (a speaker) while the speaker varies in intensity. The coupling coefficient b is determined by the strength of coupling from the intensity of the speaker, heating from the nicrhome wire, geometry of the tube and the distance between the tube and the speaker. An Arnold’s tongue is shown in Figure 2.
Figure 2. This is a typical Arnold’s tongue. The three states of the system are: Completely coupled (I), completely decoupled (III), and the transition state where it wanders (II).The difference in frequency is along the x axis and the coupling coefficient is along the y axis. The colored lines in region I show the rate of synchronization, which will be talked about more extensively later in the paper. Figure was generated with MATLAB, courtesy of adviser Wick Kurt.
Penelet and Biwa’s paper, Synchronization of a thermoacoustic oscillator by an external sound source, used an Arnold’s tongue to display their data. Starting from a frequency of 170 Hz, the system coupled for a maximum frequency difference of nearly 30 Hz [1].
The region inside the Arnold’s tongue is the coupled state which is described by a rate of synchronization between the thermoacoustic oscillator and the driving force. If the coupling is stronger, then the system will couple more quickly. [8] Describing the thermoacoustic oscillator as
and the speaker as
We can model the two nonlinear oscillators system when it’s weakly coupled of by their phase difference
If the coupling coefficient is large enough so that the inequality
is satisfied, then the thermoacoustic oscillator and the driving force will converge to the same frequency and the same phase. The parameters WOscillator and WDriving are the angular frequencies of the thermoacoustic oscillator and the driving force, respectively, and b is the coupling strength coefficient of the system. Since the system will always converge to the same phase, this suggests that the initial condition of the thermoacoustic oscillator’s phase is arbitrary. The rate of convergence to the same phase for different initial conditions can be demonstrated by the phase difference between two separate trials. We expect to see exponential convergence, so by taking the log of the difference in phases between two trials, there should be a linear relationship through time. [8]
X1(t) and X2(t) are the position of two separate trials at time t, and X1(t) and X2(t) are the initial position of each trial. rsync(t) is defined as the rate of synchronization. Since X1(t) and X2(t)become closer to each other throughout time, this should be a negative linear correlation.
For our experiment’s conditions, we found the time of synchronization of the system too short and immeasurable, instead we focused on the time of desynchronization of the system. We expect the rate of decoupling to be modeled the same way as the rate of coupling and can therefore use the same Arnold’s tongue concept. The only difference should be that there should be a positive correlation because X1(t) and X2(t) should become more different as time passes.
We expected to observe that as frequency differences between the thermoacoustic oscillator and the external source increased the time for desynchronization should exponentially decrease. This is modeled by the equation
where ∆f is the frequency difference between the thermoacoustic oscillator and the external source, foscillator is the fundamental frequency of the thermoacoustic oscillator, t is the time of desynchronization, and C is an arbitrary constant. C is included because this formula should only work for the region in the Arnold’s tongue in which the thermoacoustic oscillator and the driving force are not coupled indefinitely, so it is just to offset the data from ∆f. Altogether the frequency difference, coupling coefficient, and time axes make up a 3 dimensional Arnold’s tongue.
Synchronization demands for the fundamental frequency of the oscillator to shift and match that of the external driving source. Since the fundamental frequency of the tube depends on its length and the speed of sound, one of those parameters should have to change to create a new fundamental frequency. Because the length of the tube is fixed, it means that the speed of sound must be the parameter that changes. The speed of sound inside the oscillating tube depends on the temperature and the medium it propagates across. It is modeled by the following relation where c is the speed of sound and T is temperature. [8].
As the temperature inside the tube increases from the heated stack it eventually reaches a quasi-static state, where the temperature fluctuations are negligible, and the resonant frequency is assumed to be constant. However, we expected to find a shift in the resonant frequency of the tube when we waited for the stck to reach an equilibrium temperature. This can be shown by looking at the relation between ΔT (change in temperature) and ∆c, the speed of sound by comparing the speed of sound in two different cases for Equation 1.
Experimental Setup
Our thermoacoustic oscillator is large glass tube 50 cm in length and 5 cm in diameter. At one end of the tube we have a rigid stopper to aid in the acoustic wave and a microphone inserted in the stopper to capture the resonating wave. At the other end, we have the loudspeaker. A porous ceramic stack that houses a circularly threaded nicrhome wire on the side facing the stopper is placed inside the tube. Providing the current to heat the stack there are two copper wires coming out the right side of the tube.
Figure3. On left-Image of set up in lab with labeled components. The glass tube is raised up to resonate without damping and is centered in front of speaker for symmetric aperture. Figure 4. On right-Schematic diagram of complete experiment showing all of the equipment and its configuration
Data Collection
There were many steps in our data taking process. First, we had to make sure that the resonant frequency was the first harmonic of the thermoacoustic oscillator. We also had to confirm that the frequency stabilized, after a certain amount of time. Afterwards, we began taking data to create our Arnold’s tongue. This was done by finding the desynchronization times for different frequencies and amplitudes of the speaker. Lastly, to try and find an explanation for the desynchronization observed, we measured the temperature inside the stack.
Before we collected our data, we first had to confirm that the thermoacoustic oscillator’s frequency response function behaved how we expected it to. Specifically, we wanted to see a nonlinear oscillator with a large peak at the fundamental frequency and smaller peaks at its harmonics.
Figure 5– The amplitudes of different frequencies for the thermoacoustic oscillator were measured using a microphone and a lock-in amplifier.
Also, it had to be verified that the frequency, and therefore the temperature, reached a steady equilibrium value after an extended period of time.
Figure 6- The resonant frequency of the thermoacoustic oscillator was found to stabilize after about 1200 seconds, or 20 minutes. We used this data to determine how long to wait for the thermoacoustic oscillator to reach an equilibrium state before taking our data.
After about 1200 seconds (20 minutes) the frequency reached an equilibrium state, which corresponds to an equilibrium temperature in Equation 8.
With this information, we were able to begin taking data to test our hypothesis. We measured the time of desynchronization for points along the edge of the Arnold’s tongue. A typical graph of the phase relationship between the thermoacoustic oscillator and speaker vs. time of desynchronization is shown in Figure 7.
One thing that is immediately interesting is how the oscillator appears to couple initially, and then begins to decouple around the 3 second mark. Also, it is intriguing how the oscillator decouples in a sinusoidal fashion. This was unexpected, and it was observed in all of our datasets, however the phenomenon will not be analyzed in this paper.
Because of some peculiarities seen when analyzing our data, we also recorded the temperature inside the thermoacoustic oscillator for various situations. This will be shown in the Results & Analysis section.
Results & Analysis
Analysis and Results
Rate of Synchronization and Desynchronization
After gathering the preliminary data, trials of the same type were compared (i.e. the speaker is at 168 Hz and half of its max amplitude.) Figure 8 shows what two of these trials plotted on the same graph looks like. We expected to see a positive linear correlation when comparing time and the rate of synchronization when both trials decoupled. The latter was defined in Equation 6 as
This was done for many data sets, and Figure 8 and 9 show what was typically seen. What is peculiar is that there appears to be a negative linear correlation for the first two seconds and a positive linear correlation for the latter half of the graph. This suggests that the trials initially experienced synchronized and converged towards the same phase, but then drifted outside of the inner part of the Arnold’s tongue and began to desynchronize. This can either be because the speaker’s frequency or the thermoacoustic oscillator’s resonant frequency shifted. It is unlikely that our driving frequency changed at all, so it is probable that the resonant frequency of the thermoacoustic oscillator is what changed. This could possibly be explained by temperature fluctuations within the thermoacoustic oscillator which is discussed in further detail later in this section.
Even though there were clear trends that were apparent in all of our data, we did not have time to find the distribution of rates that occurred. Because of this, our model for synchronization and desynchronization was purely qualitative.
We wanted to observe the rate of synchronization within the Arnold’s tongue, but there were complications. Either the speaker was not loud enough to couple with the thermoacoustic oscillator, or the stack driving the thermoacoustic oscillator was “blown out” by the speaker, and the thermoacoustic oscillator stopped producing noise. This effect will be analyzed more later in the paper.
Arnold’s Tongue
To get a visual representation of how our thermoacoustic oscillator and speaker coupled in frequency, we created an Arnold’s tongue. We also added the time axis to the Arnold’s tongue, which is shown by color coding different data points for different times of decoupling.
Figure 11- This is the Arnold’s tongue obtained from our data. The inner line shows where the speaker and the thermoacoustic oscillator couple indefinitely. The outer line signifies the frequency difference at which the speaker initially couples with the thermoacoustic oscillator, but eventually decouples. The color coded dots depict the time it took to decouple.
The “warmer” colors (i.e. red, orange, yellow) show faster decoupling times and the “colder” colors (i.e. blue and green) show slower decoupling times. The black dots refer data that did not decouple after 15 minutes. The inner edge of the Arnold’s tongue indicates where the thermoacoustic oscillator and the speaker were coupled indefinitely, and the area between the outer edge and inner edge shows the region in which they initially coupled, but eventually decoupled. Outside of the outer edge, no coupling occurred. Coupling was experienced for a maximum difference in frequency of 12.9 Hz when our speaker was at its maximum volume.
The small asymmetry for the inner part of the Arnold’s tongue can be explained because the frequency spectrum emitted by the thermoacoustic oscillator may not be perfectly symmetric. The outer region was completely unexpected though. The wide range in which the thermoacoustic oscillator initially coupled and then decoupled when the speaker’s frequency was below the resonant frequency of the thermoacoustic oscillator is intriguing.
Temperature Analysis and Dependence
To try to explain this phenomenon, we measured the temperature inside the thermoacoustic oscillator while the speaker was on. If the speaker caused noticeable temperature fluctuates, it could explain why there are regions that initially couple and then decouple. Also, if the temperature fluctuations differ depending upon whether the speaker’s frequency above or below the thermoacoustic oscillator’s resonant frequency, it could explain the asymmetry in the Arnold’s tongue shown in Figure 10.
We measured the temperature along the cold side of the stack when the speaker’s frequency was above and below the frequency of the thermoacoustic oscillator. This was the most consistent and trustworthy place to measure the temperature. The hot side of the stack fluctuated by about ±10° C, far too much to get accurate data, whereas the cold side fluctuated by ±0.2° C. The results obtained were surprising.
Figures 12 (left) and 13 (right)- Figure 12 on the left illustrates the cold side of the stack’s temperature after the speaker was turned on at a frequency of 170 Hz. This is 6 Hz below the resonant frequency of the thermoacoustic oscillator. The temperature initially increases, then continually decreases. Figure 13 shows the temperature of the hot side of the stack when the speaker was turned on with a frequency of 182 Hz, 6 Hz above the resonant frequency of the thermoacoustic oscillator. In this figure, it is clear that the temperature is much more stable.
The temperature in both Figures 12 and 13 initially increases, however, the temperature stabilizes around the temperature it was at initially when the speaker has a frequency (182 Hz) above the thermoacoustic oscillator (176 Hz), but the temperature continually decreases when the speaker’s frequency (170 Hz) is lower than that of the thermoacoustic oscillator.
These results show two things. The first is that temperature is clearly affecting or being affected by the system coupling. The second is more peculiar. The temperature dependence of the system, and therefore the shape of the Arnold’s tongue, depends on whether the speaker’s frequency is higher or lower than the thermoacoustic oscillator’s frequency. This can be at least partially attributed to the major temperature changes shown above because temperature changes affect the speed of sound, therefore affecting the resonant frequency. If the resonant frequency is affected, so is the Arnold’s tongue.
We also wanted an explanation for why the thermoacoustic oscillator stopped producing sound for regions within the inner part of the Arnold’s tongue. To do this, the temperature difference between the hot and cold sides of the stack were recorded. If the temperature difference became small enough, the heat gradient would not be sufficient to create the sound in the thermoacoustic oscillator.
The temperature difference between the hot and cold sides of the stack were initially 78° C. This value dropped to as low as 40° C and stabilized around 60° C, nearly 20° C less than what was observed before the speaker was powered. This gives a clear explanation for why sound stopped being produced for the inner parts of the Arnold’s tongue.
Conclusion
The main conclusive results that were gathered from this experiment are: the thermoacoustic oscillator will quickly sync with the speaker if it’s frequency is within the Arnold’s tongue, but then it will slowly desync if its frequency is located along the Arnold’s tongue’s fringes; desynchronization times and rates of desynchronization decrease as frequency difference increases; and the speaker’s frequency and amplitude affect the temperature inside the thermoacoustic oscillator.
There were also some difficulties faced throughout our experiment. The main challenges that we had are listed below.
· The tightness of the stopper affected the amplitude of the thermoacoustic oscillator
o This affected our coupling coefficient, b
o If the stopper was too tight, our microphone would saturate
· The nichrome wire used in the stack would melt
o This was an unneeded variable which affected our coupling coefficient, b
o A lot of time was lost because we had to repair the stack
· The acoustics of the room had an effect on the resonant frequency of the thermoacoustic oscillator
o Where we stood affected the resonant frequency by as much as 0.1 Hz. This is important to realize when dealing with speaker frequencies close to the resonant frequency of the thermoacoustic oscillator.
Future Research
The speaker’s impact on the temperature inside the thermoacoustic oscillator was observed, but the reason is not fully understood. Finding how to not “blow out” the stack when the speaker’s and thermoacoustic oscillator’s frequencies are close to each other is crucial for measuring coupling rates.
Also, it is interesting that the temperature of the cold side of the stack acts so differently depending upon whether the speaker’s frequency is above or below the thermoacoustic oscillator’s resonant frequency. Understanding why this occurs better may help with preventing the speaker from “blowing out” the thermoacoustic oscillator, so this should be further investigated.
Two other interesting aspects of the experiment that are not fully explained are the opposite downward and upward trends seen in the phase relationship vs. time data, as well as the sinusoidal pattern seen when desynchronization occurs.
The first can be seen by comparing two graphs. Figure 15 (left) shows a trial when the speaker was below resonant frequency (176 Hz) of the thermoacoustic oscillator. Figure 16 (right) shows the opposite – the speaker was above the thermoacoustic oscillator’s resonant frequency. The upwards and downwards trends seen were consistent depending upon whether the speaker was above or below the thermoacoustic oscillator resonant frequency. The reason for this could be investigated more.
Figure 15 and 16- Figure 15 (left) shows the upward trend of the phase difference between the speaker and the thermoacoustic oscillator when the speaker’s frequency is less than the resonant frequency of the thermoacoustic oscillator. Figure 16 (right) shows the opposite – a downward trend when the speaker has a higher frequency than that of the thermoacoustic oscillator.
The sinusoidal behavior when the speaker and thermoacoustic oscillator decouple is difficult to see in the figures above, but it can clearly be seen in Figure 7 in the data collection section. This behavior is not intuitive because it shows that the thermoacoustic oscillator’s frequency is constantly being affected by the speaker. It would make more sense to see the phase change linearly because it would imply a completely unsynchronized system. The relationship between the decoupling speaker and thermoacoustic oscillator should be further explored.
References/External Links
2. Swift, G. W. (1988). Thermoacoustic engines. The Journal of the Acoustical Society of America,84(4), 1145-1180.
3. Spoor,P. S., & Swift, G. W. (2000). The Huygens entrainment phenomenon andthermoacoustic engines. The Journal of the Acoustical Society of America,108(2),588-599.
4. Noda, D., & Ueda, Y. (2013). A thermoacoustic oscillator powered by vaporized water and ethanol. American Journal of Physics,81(2), 124-126.
6. Swift,G., & Wollan, J. (2002). Thermoacoustics for Liquefaction of Natural Gas. LNGTechnology,21-26. Retrieved March 16, 2018.
8. Jensen, R. V., Jones, L., & Gartner, D.H. (1998). Synchronization of Randomly Driven Nonlinear Oscillators and theReliable Firing of Cortical Neurons. Computational Neuroscience,403-408.
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