The Cybernetics of a Pulse Combustor

[WebPilot]

Picking up where I left off on the last page.

Over a small range of positions for the 'knob', 2 stable limit cycles exist. As I increase it a little more, I reach a point where the curves just 'touch', so, only 1 stable limit cycle exists.

[WebPilot]

For any further 'twist' of the knob, the curves no longer 'touch'.

Thus, NO stable limit cycles exist at this setting.

[WebPilot]

For the present pulse combustor configuration, there are two stable limit cycles. Thus, there are two curves of excitation.

[WebPilot]

... and I located this 2nd limit cycle, too.



It is the 'big brother' to the little sister since he vibrates at the same frequency but at an amplitude of 1.25 psi(gauge).

Now the reader may think these amplitudes are small compared to a pulse jet and they are.

However, I am modelling a pulse combustion burner and the experimental data published reports amplitudes varying from 0.25 to 3.0 psi(gauge).

Pulsating combustion is responding to a mathematical treatment as I always knew it would.

The veil shrouding pulsating combustion in mystery now has a serious rip in its fabric.

[WebPilot]

Some unexpected confirmation of predictions made by my 'dynamic model of a strip valve' can be read here, dynamic modeling of a strip valve.

The qualitative aspects of the 'strip valve' are important to this model.

[Graham C. Williams]

Dear Forrest.
I'm wondering if the Damping Curve is itself a function of a number of variables that can change during the motors start-up process. It wouldn't surprise me at all.
Lately I've been modelling (with NUDiS) some short valveless motors. 'Short' as in The total combustion time being a longer than half the motors cycle period. Another characteristic of these motors is the production of a shock or very steep compression wave for some period of the cycle at some point in the pipe. In order to model these I have to assume an additional degree of combustion acceleration. This technique has worked well with the FWE VIII REV08 motor and produced a good correspondence between the model and the Physical motor. Sorry for the preamble.
One small motor I tested gave me unexpected results. In the first 4 cycles after the initial 'Kick' the CC Pressure increased, cycle by cycle, as expected and the combustion gas temperature increased as expected. Then the expected shock formed in the tail, over the next 6 cycles the combustion gas temperature started to drop then levelled out and stabilised at a lower figure. I've never seen this before!
Turning to Thrust. I was expecting to see an averaged (over the last 6 cycles) Thrust in the 4 to 5 Lbs region. It was near that value. On a cycle by cycle basis the Thrust told a different story. Peak Thrust corresponded to the cycle with peak combustion temperature. The Trust then dropped and stabilised again when the combustion temp stabilised in the last part of the run. The motor was now running at about 3.5Lbs. The largest contributor to this change in thrust seems to be from the change in internal momentum of the gasses!
Over all a strange tale.
Graham.

[WebPilot]

The damping parameter in this model is a function of several variables, but these variables have been modeled as constant with time. What you are saying is possible, I guess, but I've found no pressing need to investigate it, yet.

You see, there is enough complexity with the non-linear dynamics of this model's system differential equation(s). Let me explain to the reader what I mean by this.

In one of my earlier models, this is a 'phase plot' of the change of pressure with respect to time versus the pressure as the combustor runs. This is done by say, you are holding a time piece and at each second you were able to record the pressure rate and also the pressure. Then, you would plot these points (pressure, pressure rate) on a graph in temporal succession, as I have done here, and connect the points with lines.



The reader does not have the benefit of seeing this 'real time', but the plot starts in the center and goes around in a CW manner. Where it is dark, the model spins around there for a couple, or so, of cycles. However, it does not take too long before the model begins to 'jump orbit'. By this I mean the model can be on an inner orbit and then jump to an outer for a while, only to jump to some other orbit. I found it impossible to figure where it would go next! However, notice there are bands of white, where it never seems to want to go.

I named this the 'florence ribbon' when I posted it quite a while ago on the forum.

Now if you were to plot your data 3 dimensionally, say, using the points (time, pressure, pressure rate), you would get something like this:



You will get 'coils' that expand and contract apparently at random with ever increasing time.

I was beginning to wonder about this randomness and 'speed changes' in the pressure rate, so I turned off lines in my plotting program and plotted each point as a dot. I didn't quite expect this.



I call this Schwarze Sonne (ger. "Black Sun").

Look at the delicate fine structure. It's not a true fractal, since it is not self similar, but I found it 'absolutely beautiful'.

This is what I find so fascinating about non-linear dynamics. Is this behavior complex enough? I find it so.

[larry cottrill]

In an odd way, it reminds me of a tool structural engineers used to use occasionally for dynamically loaded structures such as bridges, called influence lines. There are separate influence lines for shear, tension, bending moment, etc. but what they all are is a plot of how a point on the structure undergoes changes in internal forces (e.g. unit stresses) over time (say, as a large truck enters the bridge, speeds across its length and finally departs off the other side). To a neophyte, these graphs can be quite surprising; for example, when the front wheel of a truck rolls onto the first inch of a bridge, that turns out to be an impact load! (It turns out that any force that suddenly appears is an impact load.) And the stress variation on that one inch slice of bridge can be somewhat strange as the various wheel loads of the truck "impact" it and then proceed across. (This can be done at any selected point of the structure, of course, and can be much more complex in less determinate structures; a simple bridge span is an extremely elementary example.) This was interesting to me in those days (away back in the 1970s), since this seemed to be the only way to look at the structure dynamically -- usual design methods were quite static, and based on what the engineer thought were "worst case" positions and magnitudes of loads. Of course, with highly developed computer aided design, methods available to "ordinary" practicing engineers are undoubtedly FAR more dynamic now, especially with the emphasis on seismic loading and so on.

L Cottrill

[WebPilot]

In my dynamics training, the influence coefficient matrix is the inverse of the stiffness matrix.

[WebPilot]

For this model, this is the phase relationship between Pressure and Flow. Flow lags pressure.



"Stopped organ pipe theory" was NOT used in this model. The reader is urged to realize that there are other oscillatory phenomena with similar phase characteristics.

This can be shown in a set theory diagram. Organ pipe theory is merely a subset of a much larger group of oscillatory phenomena with similar phase characteristics.

[WebPilot]

Just when I wished to refer to this thread, the host is down for page 1. I may have to make a more permanent change. For now, ...

Re-Re-restoration of links to missing images found on page 1 of this thread:

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2Mike-3a.gif

(c.t.)
cyber-1.gif

(c.t.)
cyber-2.gif

[kenshin]

Some professional research
http://www.ncbi.nlm.nih.gov/pubmed/12779686
Maintenance of chaos in a computational model of a thermal pulse combustor

Chaos. 1997 Dec;7(4):605-613.
Maintenance of chaos in a computational model of a thermal pulse combustor.

In V, Spano ML, Neff JD, Ditto WL, Daw CS, Edwards KD, Nguyen K.

Naval Surface Warfare Center, Carderock Laboratory, West Bethesda, Maryland 20817.

The dynamics of a thermal pulse combustor model are examined. It is found that, as a parameter related to the fuel flow rate is varied, the combustor will undergo a transition from periodic pulsing to chaotic pulsing to a chaotic transient leading to flameout. Results from the numerical model are compared to those obtained from a laboratory-scale thermal pulse combustor. Finally the technique of maintenance (or anticontrol) of chaos is successfully applied to the model, with the result that the operation of the combustor can be continued well into the flameout regime. (c) 1997 American Institute of Physics.

PMID: 12779686 [PubMed - as supplied by publisher]

[WebPilot]

 
 "The dynamics of a thermal pulse combustor model are examined. It is found that, as a parameter related to the fuel flow rate is varied, the combustor will undergo a transition from periodic pulsing to chaotic pulsing to a chaotic transient leading to flameout .. Finally the technique of maintenance (or anticontrol) of chaos is successfully applied to the model, with the result that the operation of the combustor can be continued well into the flameout regime."

   - "Maintenance of chaos in a computational model of a thermal pulse combustor."


Thanks for that.

[WebPilot]

☢ links verified