dynamic modeling of a strip valve

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moving forward ...

Prior to actually calculating the flow for each dfr (from 0.1 to 0.59), I've first plotted the maximum theta values as a function of driving frequency ratio.



I expected a non-linear relationship. It is sort of cheating to put a 5th order polynomial through 6 data points, but it will suffice for now.

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If you're not making mistakes, you're not learning anything.
- an old adage

I try my hardest to make my posts as error free as possible. I make mistakes, sure, but you, the reader, don't see most of them. I try to find references to support and substantiate most of my claims.

To boldly go where no man has gone before ...
- Captain James Tiberius Kirk of the starship, ENTERPRISE

This thread is about a non-linear engineering problem. I can't give you a reference because I do not have and am not using one. This is totally my own engineering, so it may be right or it may be wrong. I've corrected it once already due to it contradicting my 'engineering sense'. It wasn't a big correction and since then, it's again made pretty good 'e' sense to me. Not now.

This is the 'cutting edge' of what my model now proposes. It makes me think I have to revise it again.

However, I find it fascinating. So, with some reluctance in posting, what you are about to read in my next couple of installments may be some of the best science fiction you will ever read coming from my keyboard and mouse.

But on the flip side of the coin ... truth is sometimes stranger than fiction.

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It is possible to use a formula out of any Engineering Mechanics text to determine the theta value for a uniform beam (in this case strip), clamped at one end, and with a uniform loading (in this case gauge pressure) on one face.

I have placed this value on my previous graphic as a straight line.



There are 3 rather startling things to realize about this plot.

1. for dfr's less than about 0.36, the amplitude of the vibrating strip is less than that of the static deflection, or it is attenuated,
2. at a dfr of about 0.36, the vibrational amplitude equals that of the static deflection,
3. for dfr's greater than about 0.36, the vibrational amplitude is greater than that for the static deflection, or it is magnified.

I would be quick to dismiss all of this as poppycock and balderdash except for the following arguments

1. this is a non-linear vibration problem, so one cannot use linear vibration theory to describe it, and
2. where have I seen this dfr ratio of 0.36 before? I tried to keep this model general which is why I dismissed the modeling of a diode (as some wanted to do) or I would never have gotten this far. I also kept it general enough so it could be used in a variety of cases. Thus, it would be applicable to any strip valve vibrating between a hard face (valve seat) and free on the amplitude extreme.
   The Argus V-1 folded valve is one of those other applications. The first run of that analysis using the FE element method can be viewed here on my Mode 1/0 thread. I computed the dfr ratio there as 40 Hz / 115.6 Hz = 0.346 .

Coincidence? I don't believe in coincidences.

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It is customary in the field of vibrations to work with a dimensionless quantity known as a magnification ratio. This is obtained by dividing all thetamax's by the constant, thetastatic. Then plot again versus dfr.



Note the vertical line through dfr=0.36 .

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Knowing the effective length of my strip valve and the above relationship between thetaMAX and dfr, I can compute the maximum lift (delta) at the tip of the valve as it is forced to vibrate over a range of driving frequencies.

The tip amplitude varies from about 5/64" to 1/8" over this range.

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Recalling my previous graphic for w/wn=0.59, theta (in purple) is plotted as a function of dimensionless time and values are read on the y2 axis at the right. At tau=5.32, the valve closes properly.



In preparation for computing the "tip" speed of the valve before it "crashes" into the wall, we need to calculate and plot how the change of theta per unit time varies with time. This is elementary calculus. The change of theta per unit time is merely the slope of the theta line for each value of time.

This can clearly be seen in the following graphic. The slope of the theta line is given by the green curve.



From this plot, the slope (d theta/ d tau) of the theta line (in purple) at the end of the -tive portion of the cycle or tau = 5.28 (see vertical dashed gray line) is approximately 0.1, the value of the green line.

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The interested reader should look over again at the graphs I have posted earlier of 'valve box pressures over one cycle' for each of the following dfr 's.

Next, I took all of the data calculated for each dfr, 0.1 thru 0.59, and computed d(theta)/dtau for each at the end of the cycle. That is, I did this to all save for dfr=0.4. If you check that plot, the valve closes 'late'; it does not close until the chamber has become +tive. I computed its d(theta)/dtau value, at the time when the valve finally does close.

Here are a few words concerning terminology that I use.

1. d(theta)/dtau is known as the 'angular velocity with respect to dimensionless time'. It is expressed in units of [radians/1]. My acronym for it is 'angular velocity wrtdt'.
2. d(theta)/dt is well-known as the 'angular velocity' and it is expressed in units of [radians/sec].

Here are some additional words concerning units conversion.

1. d(theta)/dtau to d(theta)/dt, simply multiply by the natural frequency, wn, of the system.
2. radians to degrees, multiply by 180/pi

I now have 6 data points (a lot of work for only six) and have plotted them as 'Angular Velocity vs. DFR' here:



I have additionally indicated on this graph

1. the regions where the dynamic amplitude is attenuated, equal to one (at dfr =0.36) or magnified, and
2. that for dfr =0.40, the final angular velocity occurs in the red zone, or box pressure has become +tive when the valve finally closes.

[Mike Everman]

Faaaascinating, I look forward to the image. You're making me want to build a valver!

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build ... build ... build

We can see some trends in the previous graph.

1. along the interval, 0.10 < dfr < 0.3, angular velocity increases with increasing dfr
2. along the interval, 0.30 < dfr < 0.5, more data is needed
3. along the interval, 0.50 < dfr < 0.59, angular velocity again increases with increasing dfr.

Why is the determination of angular velocity important? Well, for one reason we can calculate the velocity at the tip.



The translational velocity at some location along the 'strip valve' is the product of the distance from the point of rotation to the point in question times the angular velocity. This translational velocity is a minimum at the point of rotation and a maximum at the 'tip'.

For example, let's calculate the 'tip velocity' for dfr=0.59 .

Assuming the length of the valve is 1",

1. tip velocity is 211.8 radians/sec times 1 inch giving 211.8 in/sec
2. 211.8 in/sec is 17.65 ft/sec
3. 211.8 in/sec is 12.06 miles/hr !

Considering the valve is oscillating at a couple of hundred cycles/sec, they are crashing against the plate this many times per second.

It is no wonder they fail. I am amazed they hold up for any length of time at all!

So, dfr = 0.59 is the point where the valve closes properly, but it is not the proper operating point for any kind of valve life using standard materials.

[WebPilot]

Here is some background on the relationship between angular velocity
(d theta / dt) and translational velocity. This is 1st year college physics, but you may have seen it in high school.

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I saw something in the plots of the output data from my little model that started me thinking if it was 'new' or just something I hadn't seen for a while. I started plotting sine and cosine waveforms and their multiplicatives.

You are going to see this again, so it will be worthwhile to go 'play' with your favorite plotting program.

[WebPilot]

With the preliminaries out of the way, now comes the stuff from Star Trek. Mr. Spock would raise an eyebrow.

I have added several additional points in the range where the dynamic amplitude is a little less, equal to and greater than that of the static pressure loading.



... and what to my wondering eyes should appear, but ...

1. not only is the angular velocity decreasing as you decrease the dfr from 0.59 to 0.43, but
2. the angular velocity reaches a GLOBAL MINIMUM of almost zero!

I call this phenomenum 'valve gliding'.

At the end of the -tive portion of the cycle, the valve just glides in for a soft landing at a slow, ever decreasing velocity. Not only is the tip velocity zero, but the whole valve just touches the seat with a velocity of zero.

To support this idea, here is a plot of angular velocity wrtdt (and theta) vs. dimensionless time at this dfr. Note how smoothly theta 'glides' in for a 'soft landing' against the seat. Note too, how the angular velocity smoothly decelerates at the same time. Both simultaneously and smoothly approach the value of zero at the end of the half cycle!



Here is a plot of the valve box pressures at this dfr.



Note how smooth and symmetrical p12 (pressure drop across the orifice) and p23 (pressure drop across the valve opening) are.

Of course, this needs to be verified by the experimental method ... but it appears ... through a judicious choice of dfr, strip valve damage due to impact loading can be reduced if not virtually eliminated !

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In summary,


Click here to read what's on the extreme right.

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