S15ThermoacousitOscillator

Synchronization of a Thermoacoustic Oscillator by an External Sound Source

Bron Malcolm & Alex Medved

University of Minnesota-Twin Cities

School of Physics and Astronomy

Minneapolis, MN 55455

Abstract

An experiment capable of synchronizing a thermoacoustic oscillator with an external sinusoidal sound source was designed and conducted in such a way to collect automated, reproducible data over a large period of time. The methods employed were used to observe and measure the synchronization regions of the two devices. The regions measured compare to a theoretical model obeying the Adler equation in which an oscillator weakly couples to an external force. In addition to that, a brief exploration into the synchronization of a thermoacoustic oscillator with a non-sinusoidal, periodic external force was conducted in which synchronization was observed using signals driven by both triangle and square waves. The results from this experiment have laid a solid foundation that provides direction for further investigation into the synchronization characteristics of such devices.

Introduction

This experiment was initiated by the paper by Penelet and Biwa (see attachment for copy of their paper). The goal was to synchronize a thermoacoustic oscillator, commonly known as an acoustic laser, with an external sound source. In this case, the external sound source used was a speaker. The project had several significant obstacles that had to be overcome in order to produce the results ultimately obtained, but has laid a solid foundation to build upon for future Methods of Experimental Physics students. Hopefully this wiki page will provide helpful guidelines for easier and more in depth exploration of this topic.

Theory

Acoustic Laser: as depicted in Figure 1, converts thermal energy (heat) into acoustic energy (sound), or vice versa. The former case is what was investigated in this particular experiment.

Figure 1: Heating the side of the stack facing the closed end of the resonator (tube) to high enough temperatures produces a temperature gradient along the stack that results in pressure oscillations and a sustained emission of sound.

Measurements of the synchronization between the acoustic laser and the speaker used in this experiment tested whether it obeys the Adler equation describing the case where two devices are weakly coupled.

Adler Equation:

In the Adler equation

represents the difference in phase between the speaker and the acoustic laser. When they reach synchronization will become zero and solving the Adler equation for results in a plot known as an Arnold tongue (Figure 2).

Figure 2: Arnold tongue plot as predicted by the Adler equation governing the case where two devices are weakly coupled.

Apparatus

A list of the devices used in this experiment are as follows:

Thermoacoustic Oscillator: 0.5m glass tube with 5cm diameter closed at heated end with rubber stopper, Porous ceramic stack placed 11.5cm from stopper, and Nichrome resistive wire (8Ω) to heat one end of stack

Microphone: Behringer ECM 8000 microphone with a Behringer Minimic MIC800 preamp

Signal monitored by an external oscilloscope and sent to NI DAC card read by LabVIEW program

Speaker: Neolite SW6.5 Neodymium 6.5 driver encased in a wooden case

Powered by the Agilent 33120A function generator through the MPA-200 speaker amplifier. Function generator controlled in LabVIEW program and signal is sent to NI DAC card read by LabVIEW program and monitored by an external oscilloscope.

Software: LabVIEW was used to perform automated data collection and MATLAB was used to analyze data

Figure 3 shows a block diagram that roughly outlines the experimental setup used for this project. It does not, however, provide very much detail in exactly how things were set up and therefore a better understanding of how everything was arranged in the lab can be viewed in Figure 4.

Figure 3: Block diagram of the experimental set up..

(a)

(b)

(c)

(d)

(e)

Figure 4: Five pictures taken of the apparatus. (a) provides an overview of the entire set up. Important to note that only one function generator was ultimately used to power the speaker, and only one DC power source was used to heat the stack. (b) is shows the acoustic laser from the side. The microphone is insulated and pointed at a small hole in the closed end of the tube while the open end is less than a 1 cm away from the speaker. (c) is an overhead view of (b). (d) is a close up on the closed end of the tube and how it is configured. Shown in (e) is the spiral pattern in which the nichrome resistive wire is embedded in the side of the stack.

When the speaker and acoustic laser are both operating, synchronization can be observed using an oscilloscope in the XY display format to produce Lissajous curves generated by their respective signals (Figure 5). This is a good way to gain a rough estimate for where the boundaries between synchronization and non-synchronization are located.

Figure 5: Lissajous curve of unsynchronized signals from the speaker and acoustic laser.

Data Collection

All data collection was performed using an automated program designed in LabVIEW that controlled the DC power supply and the function generator, and read the signals produced by the microphone and function generator. The signal from the function generator was read because the signal from the speaker amplifier would pick up the signal produced by the acoustic laser, which resulted in poor data. In general, data was collected by starting at the lowest speaker frequency and voltage settings (155Hz and 0.5V, respectively) and increasing the frequency in 0.1Hz increments up to 185Hz. For each increment 1 minute of data was collected. After reaching 185Hz the speaker voltage was incremented by 0.5V and the frequency was reset at 155Hz, and the process was repeated until enough data was collected.

This method can be improved upon by collecting data for longer than 1 minute because around the boundary between synchronization and non-synchronization it can take longer than a minute for the two devices to synchronize. The increment at which the voltage is adjusted could be made smaller to provide more accurate results, as well.

Data Analysis

Data analysis was primarily done in MATLAB, which could read the large amount of data collected (1 file per frequency and voltage setting producing about 3600 data files containing 720000 data points each). The information that was paid particular attention to was the time evolution of the instantaneous phase difference between the two signals being monitored. To produce this plot (Figure 6), the analytic signal of the function generator and microphone was produced using the hilbert function in MATLAB and then the unwrapped phase of the signal (executed using a MATLAB function) was taken and plotted as a function of time.

Figure 6: An array of plots like these were used to determine the regions of synchronization and non-synchronization quantitatively. Difficulty arises around the boundary where it is hard to determine quantitatively as well as qualitatively whether the two signals are synchronized (flat narrow line versus a line is flat but jumps and/or is very thick). That is where imperfect synchronization comes into play.

Results

Synchronization with a sinusoidal as well as a triangle and square wave speaker signal was observed in the lab. Using the data collected using a sinusoidal signal an Arnold tongue was plotted as viewed in Figure 7 below. The image gives motivation to further explore the regions of imperfect synchronization to obtain a more accurate measurement of the boundary between synchronization and non-synchronization. The fact that it is possible to synchronize an acoustic laser with a triangle and square wave gives motivation to perform large data collection series that can produce a comparable Arnold tongue.

Figure 7: Arnold tongue of collected data using a sinusoidal speaker signal. Plot is comparable to what is predicted by the Adler equation.

Conclusion and Future Direction

The results clearly indicate a successful method of measuring the synchronization between an acoustic laser and speaker was created and implemented. That being said, the results of this project only scratch the surface of the overall investigation into the synchronization characteristics of thermoacoustic oscillators.

A list of possible ideas for future project are as follows:

* Synchronization with non sinusoidal sound sources * Synchronization with a second thermoacoustic oscillator * Higher precision measurements at the boundary of the synchronization region * Measuring the time it takes for the two devices to reach synchronization * Investigate the change in power produced by the two signals within the synchronization region * Varying the distance the speaker is placed from the opening of the acoustic laser

Limitations to consider are as follows:

* This experiment is a particularly loud one and therefore an isolated room away from anyone who would potentially be bothered is ideal

* The combination of being loud and the fact that data needs to be collected over a very long period of time (t > week) reinforces the idea that a separate, isolated room would be ideal.

* In addition to that, the experiment is sensitive to temperature and pressure changes, therefore, performing the experiment where there is a low amount of traffic and/or controlled temperature and pressure levels would also be ideal.

Final Remarks

We would like to thank Clement Pryke and Elias Puchner for their overall facilitation of the class within which this project was conducted, as well as Peter Martin for his advising throughout the semester. Lastly, I would like to thank Kurt Wick. With out him and his guidance this project never would have had a chance.