Control Laboratory Experiments in Thermoacoustics Using the Rijke Tube

Control Laboratory Experiments in ThermoAcoustics using the Rijke Tube

Jonathan P. Epperlein, Bassam Bamieh and Karl J. Astr˚ om¨

Abstract— We report on experiments that investigate the dynamics. The advantage of the Rijke tube experiment is dynamics, identification and control of thermoacoustic phe- the ability to produce thermoacoustic instabilities without a nomena in a Rijke tube apparatus. These experiments are combustion process. Many of the identification and feedback relatively simple to construct and conduct in a typical, well- equipped undergraduate controls laboratory, yet allow for control issues involved in combustion instabilities are present the exploration of rich and coupled acoustic and thermal in the Rijke tube experiment. Thus, this experiment provides dynamics, the associated thermoacoustic instabilities, and the an easily accessible platform within which one can explore use of acoustic feedback control for their stabilization. We the myriad issues relevant to thermoacoustic instabilities and describe the apparatus construction, investigation of thermoa- their control. coustic dynamics and instabilities in both open-loop and closed- loop configurations, closed-loop identification of the underlying The present paper aims at introducing the Rijke tube as an dynamics, as well as model validation. We also summarize experimental platform to explore thermoacoustic dynamics a transcendental transfer function analysis that explains the and their control. We present an empirical investigation — underlying phenomena. These experiments are notable for the which can be easily reproduced in a controls lab — of the fact that rich thermoacoustic phenomena can be analyzed using dynamics of the Rijke tube using closed-loop identification, introductory concepts such as the frequency response and root locus, and thus can be performed and understood by controls and standard linear techniques such as root locus and Nyquist students with relatively little background in acoustics or heat criterion. It is remarkable that one can obtain rather useful transfer. and predictive models of the system with this approach. A Rijke tube can be made out of a vertical, long, narrow I. INTRODUCTION and hollow tube, typically made out of glass (Pyrex) in our The Rijke tube is an experiment that is relatively simple case for ease of visualization. Figure 1(a) shows a basic and inexpensive to build in a typical university laboratory. diagram. A heating element (typically a resistive coil) is Despite its construction simplicity, it can serve to illustrate a placed towards the lower end of a vertical, open, hollow wide variety of mathematical modeling, empirical identifica- tube. If the coil is sufficiently hot, a steady upwards flow tion, verification and feedback control techniques. As such, it of air is achieved. An increase in the power to the coil is suitable for use in both advanced undergraduate and grad- z mic uate controls laboratory courses. The Rijke tube is perhaps the simplest illustration of the phenomena of thermoacoustic instabilities. These phenomena typically occur whenever heat Open glass is released into a gas in underdamped acoustic cavities. The tube heat release can be due to combustion or solid/gas conductive and convective heat transfer. Under the right conditions, the Controller coupling between the acoustic and heat release dynamics zo in the cavity becomes unstable. This instability manifests heating coil itself as a sustained limit cycle resulting in audible, powerful pressure oscillations. Thermoacoustic instability phenomena 0 are most often encountered in combustors [1], [2], [3], speaker w where the resulting powerful pressure waves are undesirable (a) (b) due to the danger of structural damage as well as perfor- Fig. 1. (a) The Rijke tube shown with a heating element placed towards mance degradations. In this context, they are often referred the bottom (suspension mechanism for coil not shown). Upward arrow to as combustion instabilities, and are notoriously difficult indicates steady air flow caused by the coil’s heat. (b) The Rijke tube to model due to the additional complexity of combustion with microphone, speaker and feedback controller. The external signal w is used for closed-loop identification. This research is partially supported by NSF grants ECCS-0937539 and CMMI-0626170. causes an increase in the air flow, and at some critical Jonathan Epperlein is with the Department of Electrical and Computer value, the tube begins to emit a loud steady hum like Engineering, University of California, Santa Barbara, Santa Barbara, Cali- fornia 93117, USA. [email protected] a pipe organ. Proportional acoustic feedback as shown in Bassam Bamieh is with the Department of Mechanical Engineering, Figure 1(b) can make this hum disappear with an appropriate University of California, Santa Barbara, Santa Barbara, California 93117, setting of the gain. It is important to note that this is USA. [email protected] Karl J. Astr˚ om¨ is with the Department of Automatic Control, Lund not a noise cancellation scenario, but rather a stabilization University, Lund, Sweden. [email protected] problem, in that the acoustic feedback actually stabilizes the underlying thermoacoustic instability that generates the collecting data, would do. A photograph and a diagram of sound. Distinguishing between these two settings is one part this particular arrangement is shown in Figure 2. The glass of the experimental investigation. This paper is organized as follows. We first describe the basic aspects of the apparatus construction, which though relatively simple, requires some careful attention to cer- Microphone tain parameters so as to obtain an easily humming tube. PC Section III contains the basic initial observations of the with thermoacoustic instabilities in open loop, and with stabilizing DAQ Board as well as further destabilizing feedback gains. Some of the elementary acoustic physics is diagrammatically illustrated. Section IV describes the procedure and typical results of frequency-domain closed-loop empirical identification. This Coil produces an open-loop plant transfer function that can be used for model validation. Though this transfer function has Audio Amplifier been arrived at without any underlying physical modeling, it is predictive in that it explains the initial thermoacoustic instability, the proportional feedback stabilization, as well as Power Supply Speaker the high gain instabilities of the controlled system. This is done in Section V using frequency responses and root loci of the identified open-loop model. The open-loop poles of the Fig. 2. Photograph and diagram of the Rijke tube experimental apparatus. system as well as its right half plane zeros play an important role in this analysis. Finally, we include a short analysis sec- tube is vertically mounted to a rigid frame, with the heater tion (Section VI) in which a transcendental transfer function coil mounted about 1/4 of the way up from the bottom of with an infinite number of poles (representing acoustics) is the tube. The power supply is used to heat the coil. The analyzed in feedback with heat release dynamics. A root microphone is mounted on top and in the center of the tube. locus analysis shows clearly how the coupling between The microphone signal (AC coupled) is fed via the DAQ acoustics and convective heat release is the underlying cause board to Simulink, where it is recorded and multiplied with of the thermoacoustic instability. Although the analytical the variable gain. The interrogation signal (see Figure 3) is derivations of these transfer functions are beyond the scope also added there. The generated signal is then routed from of the present paper, it is included to illustrate the tight the DAQ board to the audio amplifier and to the speaker. correspondence between models derived from the underlying Our working assumption is that the the power and pre- physics, and those obtained from empirical identification. interrogation power amp speaker mic pre-amp signal II. CONSTRUCTION OF THE RIJKE TUBE APPARATUS Rijke w We describe here the particular hardware configuration Tube used in the controls laboratory at the University of Cal- ifornia at Santa Barbara (UCSB). Earlier versions of this experiment have appeared elsewhere, namely in [4] where it Fig. 3. Equivalent block diagram of the closed-loop identification setup. was specifically used in a controls laboratory. Details of our amplifier, as well as the microphone and speaker can all basic set up can be easily modified according to the user’s be described by pure proportional gains. In reality, they particular laboratory facilities. Our basic Rijke tube apparatus each have their characteristic frequency response which may used for this experiment is composed of the following main not be flat. However, in the frequency range of interest in components: this experiment (typically 50-1000 Hz) where the Rijke tube Pyrex R Glass tube, length = 4ft, internal diameter •  acoustic dynamics are dominant, we take these components 3in (A very high aspect ratio is necessary to achieve to be pure gains and regard the Rijke tube system with the thermoacoustic instability with only moderate heater this acoustic feedback as reasonably well modeled by the power.), conceptual diagram shown in Figure 3. Heater coil made from 24 gauge NiCr wire, • Simple clip-on microphone (with built-in preamplifier), III. EMPIRICAL INVESTIGATION OF THE RIJKE TUBE • Audio amplifier, • The experimental exploration of the Rijke tube begins Speaker, • with supplying enough power for the initial thermoacoustic DC or AC power supply. • instability to appear. The effects of proportional acoustic We use a DAQ board in connection with Simulink Real- feedback are then investigated through initial stabilization Time Windows Target to realize the variable control gain and then observing other instabilities at high gains. Once and collect the requisite data. This is certainly a bit overkill, a stable system is established, the closed-loop identification a simple op-amp circuit, along with some other way of can be performed to obtain a frequency response model. A root locus analysis can then explain the initial stabilization and subsequent destabilization with increasing gains. 1) Observing the thermoacoustic instability: The heater coil power supply is turned on and increased slowly. During this process one can feel the upward flow of hot air by placing the hand slightly over the tube. Most likely, the wire will also start to glow red. There is a critical heater power beyond which the tube will begin to hum loudly. The increase in sound level up to saturation occurs slowly enough (a couple of seconds) to be perceptible. If the heater power is decreased and then increased again, a slight hysteresis phenomena is observed. p v (a) (b) (c) (d)

Fig. 5. A diagram of the fundamental acoustic mode of the Rijke tube showing its “half-wave” nature. The (a) pressure and (b) velocity spatial waveforms are shown. These oscillate temporally 90o out of phase. To the right, diagrams of the corresponding motion of gas particles (ignoring the mean convective upwards ﬂow) are shown using arrows, and the corresponding pressure distribution in color. In one half cycle (c) air rushes (a) (b) into the tube causing a pressure maximum at the middle, while in the second half (d) air rushes out causing a pressure minimum at the tube’s mid point.

observe the control signal into the speakers terminals on an oscilloscope. The oscilloscope will show that the control signal decays rapidly and hovers around almost zero if a stabilizing feedback gain is used, see also Figure 4(d). In contrast, a noise canceling system would have a persistent (c) (d) non-zero control signal canceling the persistent noise. Fig. 4. Time trace of the microphone signal (a) at the onset of instability showing growth, and then saturation of the limit cycle. Linear growth on fundamental mode unstable stable higher harmonic is unstable a semilog plot (b) of the signal’s envelope confirms initial exponential 0 K growth of its amplitude. A zoomed-in picture (c) shows the periodic, but Kmin Kmax non-symmetric limit cycle behavior. With appropriate proportional feedback, the limit cycle is stabilized as this trace of the speaker’s input signal (d) Fig. 6. A depiction of the effects of proportional feedback on the Rijke shows. tube. A minimum feedback gain Kmin is necessary to stabilize the otherwise unstable fundamental mode. There is then a critical higher gain Kmax The sound frequency f is easily measured with an oscil- beyond which a higher harmonic mode (screech) of the tube becomes unstable. loscope (about 140Hz in our setup), its wavelength is found to be approximately equal to twice the length of the tube. 3) High-gain instability: Next, the gain can be increased This is consistent with a half-wavelength standing wave in until it reaches the critical higher gain value Kmax, above the tube; it is also the fundamental mode of a tube open at which a new instability is triggered and a loud screech both ends. The basic physics of that mode is illustrated in appears. A measurement of that screech frequency will Figure 5. reveal it to be an odd harmonic of the initial fundamental 2) Proportional acoustic feedback: With the control loop hum frequency. Exactly which harmonic it is will depend connected, we apply proportional feedback to the Rijke on the details of the experimental set up (in ours it is tube. If the gain is chosen appropriately, i.e. between Kmin typically the 3rd or 5th harmonic). This phenomenon is and Kmax, the humming disappears almost instantly. This however repeatable if the experimental set up is unchanged is usually an impressive demonstration of the power of (e.g. microphone, speaker and heater locations). The screech feedback. frequency is recorded as it can be predicted from a root locus Since many students who perform this experiment are not analysis of the identified system model, and therefore can be familiar with acoustics, they are often unsure as to what is used to validate that model. happening when the tube’s hum disappears. They often say that the tube’s noise has been “cancelled”, probably because IV. CLOSED-LOOP IDENTIFICATION AND MODEL of familiarity with active noise-canceling headsets. However, FITTING the process here is fundamentally different. The feedback Now we perform system identification to obtain some has stabilized the thermoacoustic instability which caused information about the dynamics of the Rijke tube. Since it the limit cycle in the first place. To verify the distinction is an unstable system, it will have to be identified while between stabilization and noise cancellation, it suffices to operating in a stabilizing closed loop. That comes with a few pitfalls, in particular simply recording the plant input noise, Schroeder-phased sinusoids [11] and sine sweeps and output and applying open-loop identification techniques, (also known as chirp signals). We experimented with all ignoring the fact that the input is the result of feedback, might three types of signals; sine sweeps, which are of use for yield wrong results; for background and more sophisticated identification of acoustic systems [12], emerged as the most methods than the one we will employ see e.g. [5], [6] and effective class, and all shown data was collected using a the references therein. sweep over the shown frequency range. Closed-loop Identification: Our method of choice is the For the identification experiment, the tube is first brought so-called indirect method: As shown in the conceptual block to a hum. Then, the feedback with a stabilizing gain is turned diagram in Figure 3, an exogenous interrogation signal is on, and the interrogation signal is added to the feedback added into the loop. The stable closed-loop transfer function signal, as shown in Figure 3. The microphone signal is T (ejk!)=Y (ejk!)/W (ejk!) can then be identified with recorded for the duration of the experiment (in our case your favorite open-loop method, and the unstable open loop about 2min), and together with the the applied interrogation can be recovered from signal forms an input-output pair, which is all the data needed KG to obtain a spectral estimate. To minimize the effects of T = 1 KG random noise, this is done several times, and an average (weighted by the covariance) of the estimated frequency as 1 T responses is formed. Figure 8 shows an averaged closed-loop G = . (1) K 1+T frequency response along with the individual experiments. This response has the signature of wave-like dynamics in K Here, is the gain we set, the gains from the other the presence of several very lightly damped modes at inte- components such as microphone preamps and speaker are ger multiples of a fundamental frequency. The fundamental assumed to be independent of frequency and absorbed into frequency corresponds very closely to the frequency of the G . The indirect method is the simplest available method to hum in the unstable Rijke tube. deal with open-loop unstable systems, and as we will see it In order to perform the least-squares fit of a finite di- works well for the present case. mensional transfer function model for T to the estimated Open-loop Identification Method: The dynamics under- frequency response, a value for the model order needs to be lying the Rijke tube are a combination of acoustics and heat selected. Figure 7 shows a 12th order transfer function fit, transfer, and are thus of relatively high order (in fact, they which nicely captures the first 6 harmonics in the frequency are infinite dimensional). Nonparametric frequency domain response. identification schemes are better suited to those types of systems than time domain based ones, since they do not force us to select a model order a priori. Instead, we will 0 directly identify the frequency response T (ej!) and then use a least-squares based method to fit a model of appro- −20 priate order over the identified frequency domain. So-called

spectral methods (see e.g. [7, Ch. 6]) estimate the frequency Magnitude (dB) −40 response as the ratio of the cross spectrum yw(!) of −60 output and interrogation signal, and the spectrum w(!) of 0 the interrogation signal. The MATLAB System Identification Identified Closed Loop 12th Order Fit Toolbox [8] offers two implementations of these methods, −360 spa and spafdr.1 Since we are expecting sharp peaks, fine frequency resolution is required, thus spafdr is the −720 Phase (deg) right choice. The least-squares fit is then performed using −1080 the FREQID Toolbox for MATLAB [9], [10]. 100 200 300 400 500 600 700 800 900 Interrogation Signal: An interrogation signal should Frequency (Hz) have rich frequency content, while due to actuator and sensor limitations in physical systems, amplitudes should Fig. 7. Closed-loop frequency response obtained by a non-parametric th be kept reasonably small. Popular choices include white spectral estimation, and a 12 order least-squares fit. Note that due to the log-scale, the seemingly large deviations in the ranges between the peaks are actually very small. 1While spa and spafdr both estimate the cross spectrum and input spectrum by by a smoothing window to what roughly amounts to the Discrete Fourier Transforms of input and output data, spa performs the To obtain the transfer function G as a parametric model of windowing in the time domain, whereas spafdr applies the window in the open loop, the fitted model T is then plugged into (1). Of the frequency domain. Since a narrow frequency domain window, which is what is required, corresponds to a wide time domain window, using spafdr course it is also possible to apply (1) to the nonparametric allows us to specify a small window, resulting in a drastic decrease in estimated frequency response at each frequency, thereby computation time compared to the large window we would have to specify obtaining a nonparametric model of the open loop. Both of to achieve the same resolution using spa. Another important distinction between spa and spafdr is that the latter allows for frequency dependent those possibilities are compared in Figure 9, they are in close resolution (hence the name), but we did not make use of this feature. agreement, which is encouraging. We note with satisfaction 0 0

−20 −20 Magnitude (dB) −40 Magnitude (dB) −40

−60 −60 0 0 (Weigthed) Average From Identified Closed Loop −720 From Fitted Closed Loop −360 −1440 −720 −2160 Phase (deg) Phase (deg)

−2880 −1080

500 1000 1500 2000 2500 100 200 300 400 500 600 700 800 900 Frequency (Hz) Frequency (Hz)

0 Fig. 9. Open-loop frequency responses, obtained by applying (1) to the identiﬁed closed-loop response at every frequency (green) or −20 to the ﬁtted closed-loop response (black)

Magnitude (dB) −40 So the standing wave corresponding to the ﬁrst peak would −60 0 be a half-wave, for the second peak it would be a full wave, (Weigthed) Average for the third one and a half waves and so on. By placing the −360 microphone at e.g. the center, we placed it where all the even numbered waveforms have a pressure node, and hence their −720 contribution is not registered by the microphone, a pressure Phase (deg) −1080 sensor.

This would not be surprising at all for a tube without a 100 200 300 400 500 600 700 800 900 Frequency (Hz) heater; that it still holds true with the heater indicates that Fig. 8. Closed-loop frequency response obtained with a sine sweep over the its effect, the thermoacoustic effect, is pulling the ﬁrst mode range of 0-2.5 kHz (top) and 0-900 Hz (bottom). The response below 20Hz, into the right half plane, but besides that, regular acoustics which is outside the audible range, and above 1kHz is likely dominated by dominate the response. microphone and speaker distortions. The range 0-1 kHz however exhibits typical wave-like dynamics with resonances occurring at multiples of the fundamental harmonic.

0 that, while the phase at the ﬁrst peak of the closed loop T o o drops by 180 indicating a stable pole, it increases by 180 −20 in the open-loop response, indicating a pole in the right half plane (RHP). Magnitude (dB) −40

Varying the Microphone Position −60

0 =1 2 So far, we have not stated, where exactly we placed ξm / ξm =3/4 the microphone during the experiments. It was not placed −360 deliberately, but rather just somewhere inside, but near the top of, the tube. The reason is that the microphone position −720 does not inﬂuence the position of the poles, hence neither Phase (deg) −1080 the peaks. However, there are special locations that do yield

100 200 300 400 500 600 700 800 900 interesting results. In Figure 10, we show the identified open Frequency (Hz) loops if we place the microphone a quarter length from the top of the tube, and in the middle of the tube. Doing so Fig. 10. Bode plots for identified and fitted open-loop responses with appears to “remove peaks,” in the former case it would be different microphone positions. Placing the microphone in the middle of the tube (⇠m =1/2) seems to remove every other peak, while placing it every fourth, and in the latter case even every even-numbered at a quarter length from the tube attenuates the fourth peak only. The very one. ugly identification data at the removed peaks, and especially at the peak This is relatively easily explained with the physical model around 660Hz, can be explained by the fact that perfect cancellation of a pole by a zero is virtually impossible; instead, one gets a pole and a zero of the transfer function, which is developed elsewhere [13], very close together – notoriously difficult to identify. but there is also a very intuitive explanation: each peak corresponds to a mode, a standing pressure wave, in the tube. V. M ODEL VALIDATION:ROOT LOCUS ANALYSIS unstable, having a positive real part. The imaginary part of We now have a model G of the open loop to validate and the fundamental mode corresponds to the hum frequency use to explain the experimental observations. We will see that heard when the tube is initially powered on. the root locus is effective in explaining why proportional The right half plane zeros explain, why instability reoccurs feedback initially stabilizes the thermoacoustic instability, at a higher gain: Figure 12(a) shows the locus and the and why a higher frequency mode becomes unstable at pole locations at the value of the gain sufficient to initially high gains. It will also give a quantitative prediction of that stabilize the fundamental mode (denoted Kmin in Figure 6). higher frequency – a prediction that can be used for model Note how all poles are in the left half plane. However, due validation. to the presence of RHP zeros, some poles will eventually be attracted into the right half plane as the gain is increased. Fig- ure 12(b) indicates that for this particular identified model, 720 it is the fifth harmonic mode that becomes unstable at higher 576 gain (denoted Kmax in Figure 6). The frequency of this

432 mode must correspond to the frequency of the screech heard

288 in the experiment as the system becomes unstable again at high feedback gains, which is indeed what we observed. 144 This serves as a useful method of model validation for this experiment. We see that the identification yielded a model that cor- Imaginary Axis [Hz] rectly predicted the instability and its frequency, and explained why upon increasing the gain, another instability occurs. It also correctly predicts the frequency of this high- gain instability. Common problems: Often, the phase of the open-loop −2500 −2000 −1500 −1000 −500 0 500 1000 Real Axis [Hz] frequency response will also drop, instead of increase, by 180o at the first peak, i.e. the open loop is identified as stable, Fig. 11. A full view of the root locus of the identified open- loop model showing the fundamental frequency at 144Hz and its while we know that the open loop must be unstable. The harmonics very close to the imaginary axis. stability of the open loop turned out to be very sensitive in particular to the amplitude and phase of T at the first peak. This is most easily explained with an argument based on the Nyquist criterion: Inspecting (1), we notice that G 720 720 has the same poles as T in negative unity feedback, so we 576 576 can assess stability of G through the Nyquist criterion. In 432 432 order for T to encircle the critical point ( 1, 0j), we need 288 288 o T > 1 and 6 T = 180 at the same frequency. Inspecting 144 144 | | Figure 8 again, we see that for the presented data, the first peak reaches only about 2dB, and the range for which it exceeds 0dB is only about 1Hz wide. Hence, if the peak is Imaginary Axis [Hz] Imaginary Axis [Hz] “cut off,” the identification will result in a stable open loop. Likely culprits are insufficient frequency resolution and too much smoothing during the spectral estimation. If increasing the resolution and decreasing the smoothing do not help, a −15 −10 −5 0 −40 −20 0 different speaker might be the solution; we found speakers Real Axis [Hz] Real Axis [Hz] to have quite different frequency responses, and some even (a) (b) added considerable phase lag. Fig. 12. Root locus of the identified open-loop model showing closed-loop It also might happen that the root locus predicts the higher pole locations at a gain which (a) just stabilizes the open-loop unstable harmonic instability incorrectly. This indicates that the initial fundamental mode, and (b) causes a higher mode to become unstable. In this particular case, that higher mode is the fifth harmonic, and its frequency closed-loop identification step was inaccurately performed must correspond to the pitch of the screech sound heard at high feedback (insufficient or noisy frequency response data, too low order gains. selection for the model fit, etc.). A repeat of the identification step with more care will typically resolve this issue and the Figures 11 and 12 show the root locus of the identified more carefully identified model will then yield the correct open-loop dynamics. The pole pattern resembles that of a prediction of the high gain instability. damped wave equation, with imaginary parts of the poles Lastly, if experiments are run for a long time, the walls of being integer multiples of a fundamental frequency, and the the tube, especially around the heater, absorb a lot of heat. real parts having successively higher damping as the mode If the identification is stopped and restarted for a new run, frequency increases. As promised, the fundamental mode is there might be no initial humming, due to the tube walls heating the air around the heater to the point where the heat 7 π transfer between air and heater is insufficient to support the 6 π π humming. In that case, one can only wait for the tube to cool 5 4 π off, or, if the setup admits, increase the power to the heater 3 π to increase the coil temperature. 2 π π VI. TRANSFER FUNCTIONS FROM PHYSICAL MODELING Im In this section we present a very brief description of a first- −π principles analysis that explains the onset of thermoacoustic −2 π −3 π instabilities using a root locus argument. The derivations −4 π of system models and underlying transfer functions will be −5 π reported elsewhere [13]. −6 π The dynamics of the Rijke tube can be succinctly de- −7 π −3 −2 −1 0 1 2 3 scribed by the block diagram in Figure 13. The transfer Re function G represents the acoustic dynamics of the tube. It is driven by both the speaker input as well as the heat release Fig. 14. Root locus plot for the heat release–acoustic velocity feedback loop G22(s)Q(s). The analytical model is utilizing dimensionless variables, with 2 from the heating coil . There is also a feedback path since the speed of sound and tube length each scaled to 1. Thus, the frequency ⇡ c/L ⇡ · acoustic velocity oscillations v(t, xo) at the heating element on the imaginary axis corresponds to a physical frequency f = 2⇡ = c 142Hz. (located at xo) modulate the convective heat release from 2L ⇡ the coil. This feedback effect is depicted by the transfer function Q. It is this coupling between acoustic dynamics effect of locating the poles of G22 slightly to the left of the and convective heat release from the coil that gives rise to imaginary axis rather than being exactly on the imaginary the thermoacoustic instability under appropriate conditions. axis (typically with higher frequency poles being shifted further into the left half plane than lower frequency modes). Microphone measurement The convective heat release dependence on velocity can be Speaker Signal G11(s) G12(s) p(t, xm) described by a memoryless nonlinearity with a square root G(s) dependence followed by a first order lag. The linearization G21(s) G22(s) of velocity fluctuations around the base velocity thus has a Heat released Velocity v(t, xo) transfer function which acts like a lowpass filter: by the coil at the coil Khr Q(s) Q(s)= . (3) ⌧hrs +1

Fig. 13. A block diagram representation of the Rijke tube dynamics Khr is an unknown, positive gain, and ⌧hr is the time constant depicting the feedback interconnection between the acoustic dynamics G, of the heat release, which for our setup is estimated (see [14] and coil heat release dynamics Q. Acoustic velocity fluctuations v(t, xo) at and again [13]) to be of the order of 1. the coil location xo modulate the convective heat release q(t) from the coil. This heat in turn acts as a driving force for the acoustics. G is a distributed Since the gain Khr is of known sign but unknown size, a transfer function which describes the acoustics of the entire tube, while Q root locus analysis is the natural choice. The root locus for is lumped since it describes local effect of velocity on heat release. the open loop G22Q is shown in Figure 14 and we see that branches originating from every odd multiple of j⇡ cross The initial humming is a manifestation of the thermoa- into the right half plane, indicating that for sufficiently large coustic instability, which is caused by the feedback between Khr, the interaction of acoustics and heat release is unstable. Q and G22 in Figure 13. The remaining transfer functions in This observation is “robust” with respect to the actual value G are not relevant to this analysis and will therefore not of ⌧hr. be considered here. The acoustic dynamics are described Since in the actual transfer function with damping terms, by the driven wave equation, and one can derive [13] the the poles at j⇡ will be closest to the imaginary axis, they ± transcendental form for G22 as can be expected to be the first to cross into the right half 1 plane, at which point the system will exhibit exponentially G (s)= . (2) 22 cosh(s/2) growing oscillations, with a frequency determined by the imaginary part of the crossing poles, until it settles into a This transfer function is irrational and has an infinite set of limit cycle. The normalized frequency ⇡ corresponds to a poles lying along the imaginary axis. We note that this form physical frequency of f 142Hz, which corresponds to the was derived for acoustics without any damping. The actual ⇡ frequency observed in the experiment (see Figure 4). system’s transfer function would include damping caused primarily by viscous friction with the tube’s walls. In the VII. CONCLUSION Rijke tube such damping is relatively small and has the net The Rijke tube is an experiment that is simple to con- 2In compressible gas dynamics, heat input acts as a source term in the duct, yet illustrates complex thermoacoustic phenomena, and acoustics equations offers a platform for the application of basic as well as advanced methods of identification and control. We believe many undergraduate controls laboratories can use it to illustrate the power of even simple proportional control, demonstrate high-gain instabilities in non-minimum phase systems, apply closed-loop and frequency domain based identification techniques, and demonstrate root locus methods and the Nyquist criterion. We touched briefly on another topic: in addition to the experimental investigation, a distributed model can be derived from first principles, and analyzed with transcendental transfer functions and infinite dimensional root loci; thus, the Rijke tube allows to contrast experimental identification on the one hand, and physical modeling on the other. This will be reported in detail in [13].

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