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The rate of change of an angle


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#1 Steelsmasher   Members   -  Reputation: 152

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Posted 30 March 2013 - 01:28 PM

Hi everyone. I am hoping someone could help me work out the rate of change of angle of an object.

 

Example1.jpg

 

Basically, I have the arrow shown above moving at a velocity. But once it enters the 'field' of the circle shown above I want it to rotate towards the circle.

The part I find difficult is making sure that once it gains contact with the circle, the angle of the arrow is precisely equal to that of the tangent of the point of contact. (Hopefully that made sense)

 

I'm not sure whether the rate of change of angle would be constant or accelerating(if so at what rate?). Anyway, I'm hoping it would result in a motion that is shown by the red line.

 

Any help is appreciated!



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#2 Waterlimon   Crossbones+   -  Reputation: 2601

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Posted 30 March 2013 - 03:36 PM

I remember reading a wikipedia article about this.

 

It stated that the rate of change in the angle would accelerate to not cause an instant force to the passangers of a train driving a curve. Or something like that.

 

Ah there it is

http://en.wikipedia.org/wiki/Track_transition_curve

->

http://en.wikipedia.org/wiki/Euler_spiral


o3o


#3 Steelsmasher   Members   -  Reputation: 152

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Posted 30 March 2013 - 04:30 PM

Thanks.

A Euler spiral definitely seems to be what I'm looking for, but I imagine I'll be spending quite some time trying to apply it.

 

If maybe you could be kind enough to figure out a formula I could use?

I want the velocity of the arrow to stay constant, but need to figure out the rate at which it rotates so it manages approach the circle in a motion forming that beautiful euler spiral.



#4 Waterlimon   Crossbones+   -  Reputation: 2601

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Posted 30 March 2013 - 04:38 PM

No sorry, i dont actually understand any of that xP


o3o


#5 Steelsmasher   Members   -  Reputation: 152

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Posted 30 March 2013 - 05:52 PM

Thanks for your help anyway.

It made me manage to find this... http://www.wikiengineer.com/Transportation/SpiralCurves

 

From what I can tell, I need to look for 'θs' (which is the total angle my arrow would have to rotate), and divide it by 'ls'(this is the length of the red line, which I believe I can adjust ). That way I get the rate at which my angle should change.

But finding the right formula is proving difficult.

 

Or maybe I'm just doing it all wrong?


Edited by Steelsmasher, 30 March 2013 - 05:52 PM.


#6 DT....   Members   -  Reputation: 487

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Posted 30 March 2013 - 09:26 PM

http://en.wikipedia.org/wiki/Centripetal_force



#7 Steelsmasher   Members   -  Reputation: 152

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Posted 31 March 2013 - 02:25 PM

http://en.wikipedia.org/wiki/Centripetal_force

Unfortunately, I don't think this would help me achieve getting my arrow at the right angle.

A Euler spiral is definitely the way to go.

 

Anyway, I've spent quite some time and I've amassed the following formulas:

 

X= (Ls/100)*(100-0.0030462*θs2)

Yc = (Ls/100)*(0.58178θs-0.000012659*θs3)

θs = Ls*Dc/200

D= 200*θs/Ls

D= 1800/pi/Rc

 

I know Xc and Yand Rc but I need Dc.

So I'm hoping I will find Dc  if I rearrange the formulas.



#8 Bacterius   Crossbones+   -  Reputation: 9089

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Posted 08 April 2013 - 07:02 AM

I'm just not sure what you mean by the "field" of the circle. What characterizes the path of the arrow before it comes into contact with the circle? Is the red curve arbitrary or can you describe it with an equation (or a physical phenomenon)? How is the arrow moving? Newtonian physics? etc..

The slowsort algorithm is a perfect illustration of the multiply and surrender paradigm, which is perhaps the single most important paradigm in the development of reluctant algorithms. The basic multiply and surrender strategy consists in replacing the problem at hand by two or more subproblems, each slightly simpler than the original, and continue multiplying subproblems and subsubproblems recursively in this fashion as long as possible. At some point the subproblems will all become so simple that their solution can no longer be postponed, and we will have to surrender. Experience shows that, in most cases, by the time this point is reached the total work will be substantially higher than what could have been wasted by a more direct approach.

 

- Pessimal Algorithms and Simplexity Analysis





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