dynamic modeling of a strip valve

[WebPilot]

Again I wish to state that at the end of the intake portion of the cycle, for a dfr value of 0,431 not only does theta equal zero but so does the angular velocity, d(theta)/dt or wrtdt. The phenomenon, I call valve glide, takes place at this time.

Now I wish to turn my attention to these two curves which I find so incredibly fascinating.



If the previous was out of a Star Trek script, what I am about to write stems from the inner mind to ...

.

[WebPilot]

My model cranks out 5141 lines of data per run, each line consisting of 6 formatted cells of requisite output. I am always vigilant for 'round-off' error and can say with some confidence now, after seeing these two curves emerge, that even analytical relationships are emerging and are not being 'smudged out'.

What do I mean by this?

Simply that I began to wonder and ask myself "What analytical function and its derivative behave like these two curves?" Especially noticeable is the feature that I call 'fillets' that occur at the beginning and end of the 1/2 cycle.

So, I began to consider my entire model as a "black box"; looking to feed it a signal and then inspect the output.

I knew the input signal; I knew the output signal.

What I didn't know was what operation the "black box" was performing, so I began to experiment with a few different operations.



These look promising; notice the fillets.

[WebPilot]

Tuned to 0.431, the valve box model is acting as a sine wave multiplier of the 3rd power!

That is, it is taking the input signal and multiplying that by itself, 2 times.

We are getting an intermediate signal of A³ sin³(w/wn tau) sent to the strip valve.

Here is my Prima facie evidence of what I am proposing.


Its 1st derivative is fitting the angular velocity data nicely, also.

[larry cottrill]

Tuned to 0.431, the valve box model is acting as a sine wave multiplier of the 3rd power!

That is, it is taking the input signal and multiplying that by itself, 2 times.

Now, that is cool. Again harkening back to my radio days, it is sometimes useful to have one or more frequency multiplier stages. They are nothing more than a resonant amplifier tuned to n times the input frequency.

Very interesting in terms of the valve action, too, of course! This is getting really nice, I think.

L Cottrill

[WebPilot]

Yes, I think it's pretty neat myself the way it is developing.



In essence, this is the mechanical model of a sinusoid multiplier.

It should be noted that two signals are output.

1. if a non-contact optical displacement measuring device is directed at the valve strip, theta versus time can be measured.
2. if a laser velocimeter is directed at the strip valve, velocity versus time at that point can be measured.

It should be realized that the strip valve acts as a differentiator, that is, it differentiates its position vs. time, to produce velocity vs. time.

[larry cottrill]

Sadly, I cannot set up differentiation with my old Heathkit Educational Analog Computer -- the design of the op amps is inadequate, and they are supposedly unstable in a differentiation mode (i.e. capacitance on an input node). Modern op amp microcircuits would work fine, though. The diagram you show above could be modeled directly with a trivially small setup (you may recall that analog computers involve setups, not "programs"). All you would need besides the op amps and power supplies is a batch of precision capacitors and resistors, and of course, an appropriate signal source at the front end.

I'd love to be able to do junk like that; unfortunately, in the case of many problems, my poor knowledge of calculus would stop me from developing a proper setup. I used to have a book that had precise integrals for modelling all sorts of physical entities (like generators and motors and so on). The action of any simple physical structural element is easily handled, e.g. a loaded beam in a building can be modelled very directly, and the fun part is you can pick off outputs at each op amp, so the first one shows you the shear diagram, the second one shows you the moment diagram (integrated shear diagram), and the fourth one shows you the elastic curve (I think that's it -- what I mean is the actual deflected shape of the beam under load, i.e. the moment diagram integrated twice). People today think of these machines as crude, but you really can't appreciate what they will do until you've seen them in action. Just being able to alter one of the physical parameters by turning a knob is an amazing experience, in many cases. Of course, you also need a good signal source that can closely synthesize any desired wave form -- a "function generator", which can be pretty expensive in the case of a really good one. In many cases, a music synthesizer would work for the input. The most typical output device is an ordinary analog oscilloscope. It would be way cool to use one to work out the optimal airfoil for a given speed and loading, but you'd really have to know what you were doing.

Anyway, you do all that the old-fashioned way, i.e. mentally. And then present it very nicely, though I can't always follow the process with real comprehension. The outcome in this case is very understandable, though, even to me, and I thank you for it. Very neat indeed!

L Cottrill

[WebPilot]

This plot shows how well the analytical function is approximating the values numerically generated by the code for my dynamic model. It's doing a really good job for just my 'eye balling' it on a graph. In this case, since there is no exact solution due to non-linearity of the system equations governing the motion, the numerical model is the closest to the 'correct' answer.

There is an added parameter on the graph, and that is angular acceleration. This is related to the force acting on the strip valve, and is shown above equal to zero at the beginning and at the end of the -tive pressure cycle (and when the valve closes).

When there is NO acceleration ... there is NO force.

F = m × a

[Graham C. Williams]

Wow, That's a good fit. The second derivative looks very good.
I've been wondering if an FFT Analysis of the curves could generate a better fit? Perhaps by creating the curves as abstract amplitude values against dimensionless time then expressing the result of the analysis as a polynomial? Hmm, now I'm not sure it would be any better, the result would probably be a lot harder to work with.

Graham.

[WebPilot]

If I rewrite theta=a*sin^n (w/wn * tau), setting

1. n =2 makes the theta curve, too broad
2. n=4 makes the theta curve, too peaky
3. n=3 makes the curve as I have shown.

I can analytically estimate through inspection of the governing system equations, a value for a of 0.113 . Now if I search the numerical solution's output data file and locate the minimum value for theta, I find 0.1213783 . Setting this value for a, I obtain



which is a substantial improvement for theta, d(theta)/dtau and d2(theta)dtau2 - with the latter, the angular acceleration, still off a bit. However, without benefit of the numerical data, I presently cannot justify this modification. So, I consider this modification unwarranted or a form of cheating.

Yes, I could find a better value for n than 3, but as you speculate, and I agree, it will make the analysis more cumbersome. As it is right now, for n=3, I can use trigonometric identities to prove a couple of points.

Hopefully.

[Graham C. Williams]

Agreed.
And finally do you think that the 3rd derivative (Jerk?) in the region near where the valve opens and closes would materially influence the valve life?

[WebPilot]

Your concern is illusory.

moving forward ...

I would have liked to have posted this earlier, but I decided I needed to re-derive a formula first. I'm still in the process, but I have some early results.

I have done a Fourier analysis on the above pulse; that is, for theta. In order to verify the analysis, I decided to re-construct theta, using only 6 terms, presently.

You can see the overall shape, has begun to materialize.

[WebPilot]

Here is a 10 term approximation. Notice after 3w, the odd terms are missing.



PS
Lest we forget on this day, the historical "Day that shall live in Infamy",
December 7, 1941, the day of the surprise attack by the Japanese military on Pearl Harbor, Hawaii.

It's not in the news much, any more.

[WebPilot]

The previous two graphics are in error. I could not eliminate the lower, right side hump even if I added many more terms. It was a lot of work; resulting in no net gain. However, it did prepare me for what was to come.

The previous formula is correct, but not for this application. So, I spent part of the day coming up with a new formulation. This one is good; I can almost duplicate the signal with only 4 terms and the dc bias level.



Again, there is only one odd harmonic; namely, 3w. w is the fundamental. The remaining harmonics that I used to build the signal are even. The others that I could include, but have not shown, are all even harmonics.

[WebPilot]

Here is a glimpse of the refinement of the signal reconstruction taking place as I add two more even harmonics.

[Graham C. Williams]

Hi Forrest.
That impresses the hell out of me. Only one odd harmonic and DC, Hmm.
Is the function reasonably easy to use?

Graham.