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| Roger's diagram with numbered items |
Questions:
1. I've labeled the walls 1, 4, 6 and 13. Which of them can move? Are any of them tied together?
2. Why is there a gap at 5 between walls 4 and 6? Are they the same wall? Can one move in relation to the other?
3. I've labeled what I think are dashpots 11 and 7. Am I correct? Or is 7 an actuator of some kind?
4. Are springs 10 and 11 the same stiffness?
5. Does anything in this diagram have any mass?
6. There are circular structures at 2, 3, 8, 9 and 16. Are they the same? Are some of them joints?
7. Are your scales A and B fixed to something that doesn't move?
8. You have double headed arrows at 12, 14 and 15 indicating a freedom of movement in that direction I guess. True? Is there any significance to them being different sizes?
This diagram came from Roger Sparks on a page he put up here.
As I mentioned in your comments section Roger, my simple electrical circuit intuition is better, so I came up with a pretty simple 1st order LR circuit that I think has more than enough flexibility to model SIM's state and at least one of it's measurements. See what you think. I didn't fill in all the blanks, but I got a start. I don't know how your circuit intuition is, but it's a pretty simple circuit. The diagram is certainly simple enough. You could certainly make it even simpler and just model SIM's state (H) or the state and a measurement which doesn't behave like any of SIM's measurements precisely, but the circuit I present there allows a perfect representation of a state and a measurement. Of course we could do an RC circuit instead (which is probably more practical). If you're a HAM you maybe have all the parts you need to build a simple circuit. Do you have an oscilloscope? A signal generator? Soldiering iron?
Hi Tom,
ReplyDeleteLet me give you the hard answer first, because it is a problem--maybe without solution. The rest are easy (I think).
Items 7 and 11 are actuators. This is a problem because they require outside decision making. Something would start them moving. That is the problem and it is the same problem as moving a wall. What starts it?
OK, more detail. Your original reference called 7 and 11 dashpots which were restrictive devices, only displaying force when motion occurred. They popped a vision of hydraulic cylinder into my mind, and hydraulic cylinders are common devices in heavier machines for moving parts.
Hydraulic cylinders require pumps, reservoirs, connecting tubing, and most important, valves for control. For sure a hydraulic system would be a storage device. It would have mass but could also be air with little mass, but then you would have compressability and yet another storage system.
However, I was only thinking of the actuators as symbols of decision making. I intended for the springs to be the analogy for money storage.
That was the hard question.
The remaining questions are easy. I will answer in caps to keep them readable.
1. I've labeled the walls 1, 4, 6 and 13. Which of them can move? Are any of them tied together?
ANS: 1 IS FIXED, 4,5,AND 6 MOVE TOGETHER.
2. Why is there a gap at 5 between walls 4 and 6? Are they the same wall? Can one move in relation to the other?
ANS: THEY ARE THE SAME WALL. SLOPPY DRAWING HERE.
3. I've labeled what I think are dashpots 11 and 7. Am I correct? Or is 7 an actuator of some kind?
ANS: i WAS THINKING OF BOTH AS ACTUATORS. THEY ARE AN ANALOGY FOR DECISION MAKERS. MAYBE A POOR ANALOGY.
4. Are springs 10 and 11 the same stiffness?
ANS: NO. I WAS TRYING TO STRESS THE UNEQUAL NATURE OF THE TWO SECTORS.
5. Does anything in this diagram have any mass?
ANS: MY INTENT WAS NO. MOVING MASS IS ANOTHER STORAGE SYSTEM.
6. There are circular structures at 2, 3, 8, 9 and 16. Are they the same? Are some of them joints?
ANS: YES, THEY ARE JOINTS. I THOUGHT YOUR KEEN EYE NEEDED A JOINT, JUST LIKE THE REAL WORLD WOULD
7. Are your scales A and B fixed to something that doesn't move?
ANS: I WAS THINKING THE SCALES WERE TIED TO A REFERENCE PLANE THAT WAS FIXED TO THE LEFT WALL.
8. You have double headed arrows at 12, 14 and 15 indicating a freedom of movement in that direction I guess. True? Is there any significance to them being different sizes?
ANS: TRUE. THE DIFFERENT SIZES WAS ANOTHER ATTEMPT TO EMPHASIZE THE DIFFERENCES BETWEEN THE SECTORS.
Yep, the problem question was about the actuators. They are intended to be analogies for decision makers. It is easier to spend money than to turn a valve! LOL
I will take a look at the circuit tomorrow. the Ldi/dt doesn't scare me too much. I go to the books if I really want a measurable answer.
Thanks Roger. Decision makers *could* make it more complicated. Why not write down some equations for how the system behaves now?
DeleteThis comment has been removed by the author.
ReplyDeleteGood Morning Tom,
ReplyDeleteI see that, for some unknown reason, my comment was posted twice. I deleted one. That has happened a few times over the last two days.
I was thinking about decision making (and how to model it) last night. I think a "filter" is a decision maker. It makes a decision based on frequency. A simple coax cable is a filter when it is not a single cable, but is two cables of different impedance connected in series. The impedance difference will cause different currents to flow in each line for different signal frequencies.
So when I awoke, I took a look at your circuit. You are writing about the same thing!!!
Here is the difference: Your text and math are discussing distinct breaks in current level based on the ratios of resistors. Of course this can be done. Easier is to change the frequency. There is a logarithmic curve (to be found) describing the frequency-sensitive current distribution/division between the resistive paths.
Your circuit brought to mind that Fourier described an impulse as a infinite series of harmonically related frequencies, which I did-not/do-not understand. But, in light of your circuit, and thinking of decision making, Fourier's observation suddenly makes sense. (I won't explain how, it's complicated.)
As I read through your circuit, I wondered why you were focusing on discrete ratios between states? Because the circuit is frequency sensitive, an infinite number of solutions is possible. Why focus on a series (which brought to mind the Fourier description) of solutions?
Am I thinking the same as you? I could go on with these analogies using other circuits. But I think in a spectrum-of-frequencies way, not a harmonically-related-series-of-frequencies way. I guess I could change or learn a new trick. LOL
Knowing the system's pole tells us it's frequency response. Indeed it is a low pass filter. I could sketch it's frequency response by hand. The corner frequency will be at $w = a$ where $w$ is in radians / sec. At that point the magnitude (in terms of power) will be down 3 dB, and it will then bend down and drop at -20 dB/decade (6 dB per octave) after than. The phase will be flat out to $w = a/10$, and then will decrease at -pi/4 radians per decade until the phase changes by -pi/2, and it will flatten out there at -pi/2.
DeleteHere's the Bode plot for a low pass system. My system's response will look just like that if we multiply the x-axis by $a/100$. Now of course I'm talking about the response of the state $i_1$. The other two currents will be slightly more complicated because of the non-zero $d$ each has: that will cause the system to have a zero (not just a pole) in their transfer functions. So we'd have to find the location of that zero in both cases (not hard), and the plots will have some different bends in them (like these).
The zeros in the output transfer functions (all four of them in the case of SIM and both of them in my circuit) will cause their magnitude Bode plots to stop descending at 20 dB / decade at some frequency, at which point the magnitude will flatten out as frequency increases further. That's the effect of a non-zero D matrix: even if the contribution from the scaled state terms (Cm*H) falls to zero, they still have that pass-through term to give them a non-zero output. (Cm is my state space C matrix... I added the "m" to distinguish it from the unfortunately named measurement C).
Delete... and Good morning to you too!
Delete