Strange Wolfram|Alpha Output
Posted: 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] dθ]<sup>-1/2</sup> dθ = 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> dθ
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.
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] dθ]<sup>-1/2</sup> dθ = 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> dθ
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.
