Modeling a Driven Thermoacoustic Oscillator with the Adler Equation

Mikhail Schee & Josef Steiner

University of Minnesota

Physics and Astronomy Department

Abstract

A thermoacoustic oscillator was driven by a loudspeaker. Various aspects related to the synchronization of the system were measured and compared to a model based on the Adler equation. Initial results were unable to be reproduced, this lead to finding a correlation between the locking range and temperature. Further experimentation is required to confirm the Adler equation's ability to describe the thermoacoustic oscillator system.

Introduction

Consider a system of two pendulums hung from the same location oscillating at slightly different frequencies; their mutual influence will cause both to change in frequency. Given sufficient time, the two pendulums will come to have the same frequency and phase. This is known as synchronization [1]. This phenomenon extends to many areas of scientific research including lasers in physics, phase locked synchronization in inner ear hair cells [2], and even in clapping as an audience member in everyday life [3].

A previous MXP project measured an Arnold tongue, one consequence of the Adler equation, for a thermoacoustic oscillator being driven by a loudspeaker [4]. The goal of our project was to model other aspects of the system using the Adler equation. It is an interesting connection between sound waves and a temperature gradient which applies to thermoacoustic engines and refrigerators.

Theory

Establishing a temperature gradient across the stack inside the tube creates a standing acoustic wave in the oscillator as depicted by the red curve in the figure below.

This gives the fundamental frequency of this wave in terms of the speed of sound in air (c) and the length of the tube (L)

The Adler Equation

The Adler equation describes the change in phase difference between an oscillator and its driving source with respect to time. Initially derived for circuits, it has been modified to model mechanical systems. [5]

where ∆φ is the phase difference between the oscillator and the driving source, f is the frequency of the driving source, f0 is the natural frequency of the oscillator, Vs /V0 is the ratio of voltages of the output of the driving source and the oscillator, and Q is a quality factor which is defined to be “the ratio of energy preserved to energy dissipated per cycle in the oscillator” [6]. Q is assumed to be a constant inherent the system.

When f is close to f0, the oscillator will adjust its frequency and phase to synchronize with the driving source. When in sync, ∆φ is constant which simplifies the Adler equation. A unitless constant k is defined below and the simplified Adler equation was solved for the sine of the phase difference.

From the above relation, a range of values of f where synchronization is expected can be found. This is called the locking range:

Temperature Dependence

The slight upward slope of the initial results were attributed to the heating of the stack, which heats the air and increases the speed of sound (c) as predicted by the following relation:

where γ is the ratio of specific heats (1.4 for air at STP), R is the gas constant (286 m2/s2K for air), and T is the absolute temperature of the air in Kelvin [7].

This gives a more precise definition for f0:

As c increases, so does f0 which in turn increases the ∆φ expected when in sync.

Experimental Setup

The lock-in amplifier compared the phase difference between the thermoacoustic oscillator and the speaker. The phase data from the oscillator was measured by a microphone placed through the closed end and the phase data from the speaker was taken from the function generator. A complete seal of the tube was necessary to obtain the thermoacoustic effect, therefore the nichrome wire heating element was fed through the open end of the tube. A LabView program was used to automate the data taking process.

Results

The oscillator was driven by 10 f values near f0. The initial fluctuation, constant Δφ after synchronization, and final Δφ decreasing with increasing f are just as predicted. At f = f0, it was expected that ∆φ=0, but a phase offset was inherent in our setup. Averaging over times after synchronization, the unitless value of k was found to be (3.525 ± 0.001)*10-3.

When driven at a constant f, the increasing f0 moves the locking range causing the oscillator to go into and then out of sync with the speaker. This is exactly what was found in the plot below on the left. Below on the right is the temperature at various locations in the oscillator.

The f vs. t and ∆φ vs. t data were collected on different days, but are overlaid for comparison purposes.

The temperature inside the tube was recorded during the same trial as when the frequency was recorded. The correlation plot below shows those two plotted agains one another and has an R2 = 0.715.

This plot was fit to the relation between the fundamental frequency and temperature stated earlier and compared to the theoretical result:

The fit result was 6 standard deviations from the theoretical.

Conclusions

The initial results matched the theoretical predictions, but we were unable to reproduce them even when more tests were run under the same conditions. Investigating this inconsistency lead to finding a connection between the locking range and temperature. The promising initial results and the correlation between temperature and frequency motivate the further exploration of the Adler equation to accurately model this system.

Future Directions

• A closer investigation of the relationship between fundamental frequency of the oscillator and temperature.

• Removing the temperature dependence in the phase difference data, either by altering the setup of the heating element to maintain a constant temperature or by adjusting the phase difference data with a determined frequency and temperature relation after collection.

• A quantitative look into how the placement of the stack affects the oscillator.

• Examining the temperature dependence at more places in the oscillator, possibly using a thermal camera.

Limitations

• The thermoacoustic oscillator creates fairly high noise levels when running.

• The system seems to be sensitive to sudden changes in air pressure. We noticed that moving quickly near the oscillator would make the sound die off.

• It is important to have a tight seal on the closed end. We noticed that even running wires through the seal of stopper caused the oscillator to cease making noise.

• Measurements of frequency were sensitive to ambient noise. We noticed that talking nearby caused significant fluctuations in the readings as can be seen throughout the last three minutes of the phase difference and frequency vs. time plot.

Acknowledgements

We would like to thank Kurt Wick for his help and guidance on this project.

References

• • [1] Penelet, G. and T. Biwa, “Synchronization of a Thermoacoustic Oscillator by an External Sound Source,” Am. J. Phys. 81, 289-297, 2013

  • [2] Shlomovitz R, Roongthumskul Y, Ji S, Bozovic D, Bruinsma R. “Phase-locked spiking of inner ear hair cells and the driven noisy Adler equation.” Interface Focus,2014

  • [3] Néda, Z., E. Ravasz, Y. Brechet, T. Vicsek, and A-L Barabsi. “The Sound of ManyHands Clapping.” Nature 403 (2000): 849-50. Print.

  • [4] Malcom, B. and A. Medved, “Synchronization of a Thermoacoustic Oscillator by anExternal Sound Source.” University of Minnesota, School of Physics and Astronomy,May 2015

  • [5] Adler, Robert. “A Study of Locking Phenomena in Oscillators,” Proceedings of the I.R.E. and Waves and Electrons, 351-357, 1946

  • [6] Bhansali, P. and J. Roychowdhury, “Gen-Adler: The Generalized Adler’s Equationof Injection Locking Analysis in Oscillators,” Proc. IEEE Custom Integr. CircuitsConf., pp. 522-527, Sep. 2005

  • [7] Hall, N. “Speed of Sound.” National Aeronautics and Space Administration. 5 May 2015.