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My Technic math puzzle of the day - N. W. Perry - 2021-10-07

I haven't posted one of these for a little while (because I've learned so much already from previous ones), but here's my latest Technic math puzzle, from the 8880 Supercar:

Consider the suspension assembly pictured. The blue control arm rotates around A, and the length of AB is 60. The gray liftarm rotates around C, and of course CD is also 60 (as is AC, for that matter). The radius around D is 9.

The shock absorber is connected at B and E. Also at B, there is a gray stopper piece, and this is locked together with the shock absorber to the axle at B.

The purpose of the stopper is to wedge the liftarm between it and the control arm. When this happens, point D would be 19 LDU (its radius of 9 plus half the width of the adjacent parts) along both the Y-axis of the control arm and the Z-axis of the stopper. This means that the control arm can't be raised any more, so there must be a minimum possible value for the angle at A. That's what I want to find.
   
Right now I don't have enough values to solve the problem, because there are multiple possible solutions where D is 19 from both vectors. But eventually the length of the liftarm stops the assembly from closing any further, and I haven't worked out how to express this mathematically. I've got it "close enough" for now, with an approximate angle at A of 100.5°. But as always, I'm keen to know the precise answer, and the method to obtain it!


RE: My Technic math puzzle of the day - Orion Pobursky - 2021-10-07

I evidently solved this problem with the 8880 in the OMR. Maybe you can look at that for clues?


RE: My Technic math puzzle of the day - N. W. Perry - 2021-10-07

(2021-10-07, 2:02)Orion Pobursky Wrote: I evidently solved this problem with the 8880 in the OMR. Maybe you can look at that for clues?

The OMR model keeps the control arms horizontal, so there's some collision between these parts. (For whatever reason, the IRL set has these parts to prevent just that arrangement—perhaps because they hadn't come out with the stiff shock absorbers yet?)


RE: My Technic math puzzle of the day - Orion Pobursky - 2021-10-07

Really? I'm usually super diligent about avoiding collisions at all costs. This is why there's still an 8448 variant I haven't finished since the math gave me headaches.


RE: My Technic math puzzle of the day - N. W. Perry - 2021-10-07

(2021-10-07, 2:28)Orion Pobursky Wrote: Really? I'm usually super diligent about avoiding collisions at all costs. This is why there's still an 8448 variant I haven't finished since the math gave me headaches.

Just a bit smushed, yeah:
   

I mean you'd never notice unless you went looking for it. Only thing I guess is that the real model can't be in this pose—but then again, I think a lot of people modded their kits so that it could be! Cool


RE: My Technic math puzzle of the day - Philippe Hurbain - 2021-10-07

(2021-10-07, 1:52)N. W. Perry Wrote: The radius around D is 9.
Note that IRL the radius of "modern" beam ends is closer to 9.2 ldu. For this old design, it's even larger, around 9.8 ldu.


RE: My Technic math puzzle of the day - N. W. Perry - 2021-10-07

(2021-10-07, 12:10)Philippe Hurbain Wrote: Note that IRL the radius of "modern" beam ends is closer to 9.2 ldu. For this old design, it's even larger, around 9.8 ldu.

Which would make the minimum angle of A even greater, no? But for my purposes it's fine to go by the modeled radius. Shouldn't change the math (whatever that is).


RE: My Technic math puzzle of the day - Travis Cobbs - 2021-10-07

As far as I can tell when looking at the model IRL, the fact that the gray liftarm has divots in the side is critical to this working. The gray stopper piece that it slides by seems to only clear because of those divots. (It's hard to see for sure in the fully assembled model, and I'm not going to take mine apart to look.)

Note: I realize this doesn't answer your math question, but it changes the pertinent geometry that you're trying to figure out.


RE: My Technic math puzzle of the day - N. W. Perry - 2021-10-08

(2021-10-07, 23:40)Travis Cobbs Wrote: As far as I can tell when looking at the model IRL, the fact that the gray liftarm has divots in the side is critical to this working. The gray stopper piece that it slides by seems to only clear because of those divots. (It's hard to see for sure in the fully assembled model, and I'm not going to take mine apart to look.)

Note: I realize this doesn't answer your math question, but it changes the pertinent geometry that you're trying to figure out.

You're right, I noticed that even in the LDraw model, the stopper does actually notch into this divot. I figured (perhaps wrongly) that if I could work out the basic math, I could make adjustments to account for this.


RE: My Technic math puzzle of the day - Max Martin Richter - 2021-10-08

(2021-10-07, 2:28)Orion Pobursky Wrote: Really? I'm usually super diligent about avoiding collisions at all costs. This is why there's still an 8448 variant I haven't finished since the math gave me headaches.

Same on my side. I never got the steering shaft with its universal joint calculated in the right way.
All rest of this car is finished.