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Strange Wolfram|Alpha Output

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Strange Wolfram|Alpha Output

Unread postAuthor: DYI » Sun Jul 11, 2010 1:51 pm

Warning: this topic may contain mathematics, physics, and/or byproducts thereof. Those with an adverse reaction to such are advised to leave immediately.





I've been working on an equation to determine the opening time of a ball valve which is actuated by a constant force, pulling in a constant direction. As such, the torque and thus the angular acceleration of the handle varies over time.

It was simple to construct the equation d<sup>2</sup>θ/dt<sup>2</sup> = (rF/I)*(sin[θ + B]).

Where B is the starting angle between the (pulling) force and the handle, r is the length of the handle (the part "behind" the axis of rotation is ignored here, but not difficult to include), I is its inertial moment, and θ is the angle it makes with its initial position (fully open would be a quarter turn, or θ = pi/2).

Solving the equation, however, is rather more difficult. As a second order autonomous equation, an implicit solution is

±∫[c<sub>1</sub> + (2rF/I)∫sin[θ + B] ]<sup>-1/2</sup> = c<sub>2</sub> + t

This simplifies to:

t + c<sub>2</sub> = ±∫((-2rF/I)cos[θ + B] + c<sub>3</sub>)<sup>-1/2</sup>

Which isn't really very simple at all. Here's where Wolfram|Alpha comes in. In the output for the solution to this integral, I'm getting a disconcerting notation/error in three places.
Two of them are "1. ", and the last is "0. b" (where b is a constant). If last one is actually 0.0 * b, then an entire term in the solution disappears. As the page for the output has square brackets in the address, I'm not sure how to link to it directly. It shouldn't be too hard for the reader to enter it for himself, however. Any input as to what the issue is here, and how I should interpret that strange notation, would be greatly appreciated.

Other input on the topic (please stick to the math) that is at least somewhat informative is also welcome.
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Unread postAuthor: btrettel » Sun Jul 11, 2010 3:39 pm

While I know nothing about Wolfram|Alpha, I do know that the solution to this equation involves some atypical functions. When an analytical solution exists, but I can't really use it with a spreadsheet, I use numerical solutions. I suggest you try finding a numerical solution here.
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Unread postAuthor: lozz08 » Sun Jul 11, 2010 3:43 pm

I'm imagining this as like if you were pulling on a piece of string attached to the ball valve handle at a fixed point somewhere on the circle of the handle, and you are pulling in a constant direction.

I'm no maths genius but I've done first year maths, couldn't you just find the initial torque and the final torque, and use that to find the average torque, and then just solve for time? Sounds a lot less annoying than the integral. Sorry I can't help with the actual integral but it's a bit over my head tbh.

Also like what was said above, It might be a big help to plot a bunch of graphs from numerical inputs and perhaps you will gleen something about the problem from that.
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Unread postAuthor: Technician1002 » Sun Jul 11, 2010 5:37 pm

Don't forget to include the mass of the item providing the force. This mass if a spring will accelerate 1/2 it's mass. This applies to either a stretched spring providing linear force, or a coil providing rotational torque.
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Last edited by Technician1002 on Mon Jul 19, 2010 11:38 pm, edited 1 time in total.
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Unread postAuthor: PVC Arsenal 17 » Sun Jul 11, 2010 8:17 pm

This level math is way over my head. I haven't even touched this sort of material yet, but here is a "solution" I found when I took a crack at it using what I do know. (Calc 1)

[(I*ln2)/(rF)]*[2cos([θ+B]/2) - 2sin([θ+B]/2)] = t

I dropped the constants along the way to make it simpler.

I'm sure that's horribly wrong and of no use to you, but that's all I can offer :roll:

The most I've really done with Wolfram is use it to convert typed inputs into Math Print (fractions, exponents, etc.) so I can turn in neater work. I have heard it gets bogged down on more complicated problems.
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Unread postAuthor: DYI » Mon Jul 19, 2010 11:23 pm

Whatever that output meant, it's clear that the analytical solution does involve some nasty functions, no matter how you slice it. Having given up on an analytical solution to the problem, I got around to writing a little Flash app to solve it. Luckily, this kind of problem really lends itself to numerical solutions.

http://www.4shared.com/file/AnS1Fp1Y/ValveOpenSim.html
http://www.4shared.com/video/vDqZJO0d/ValveOpenSim.html

The application, and its .fla counterpart. Not particularly user friendly. If the force is imagined as pulling the handle, the "angle" entry should correspond to the angle between the handle's starting position, and the cable pulling it. All measurements in SI base units, and all angles in radians.
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