Modelling a carriage of a Ferris Wheel.

whoknows

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Dec 1, 2006
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My problem:

I am trying to find a model, most likely parametric, of the bottom of a swinging carriage of a Ferris wheel. Imagine someone riding in such a carriage swinging back and fourth, creating a pendulum like effect in addition to the rotation of the actual wheel. I am trying to include the rotation of the wheel (cycles/minute), the maximum angle of the swing (the angle from a vertical dropped from the point generated by the swing) and the speed of the swinging (cycles/minute).

What I have:

I have drawn two circles, a large circle being the actual wheel and another that would be traced out if the carriage would only hang without swinging (basically just a copy of the previous circle moved down the number of units that the bottom of the carriage is away from the top of the Ferris wheel). I have x= r cos x ; y= r sin x for the Ferris wheel, but I got stuck there. Then I tried x² + y² = (10)², say, for the wheel and
x² + (y + 2)² = (10)² for the traced circle of the bottom of the carriage, then set x = t to change it to parametric form and solve for y, only I cannot relate the motion of the bottom of the carriage with the motion of the wheel itself, let alone the swinging motion of the carriage. I will describe my drawing. O is the center of the wheel, O’ is the center of the circle generated by the bottom of the carriage as it goes around the wheel. P is the point of the bottom middle of the carriage.I tried drawing a line from both O and O’ to P when the point is at, say, 45 deg in relation to O. I then introduced angles theta and gamma; theta for the angle between OP and the x-axis, gamma the angle from the y-axis and O’P. That’s all I could get.

Sorry I couldn’t actually get a picture. I have no idea where to go from here. I would appreciate anyone’s help on this one, as I don’t even have my name and date on my paper yet. Thank you!
 
Hello, whoknows!

Here's a start . . .


I am trying to find a model, most likely parametric,
of the bottom of a swinging carriage of a Ferris wheel.
Imagine someone riding in such a carriage swinging back and fourth,
creating a pendulum like effect in addition to the rotation of the actual wheel.

I am trying to include the rotation of the wheel (cycles/minute),
the maximum angle of the swing (the angle from a vertical dropped
from the point generated by the swing)
and the speed of the swinging (cycles/minute).

I placed the center of the wheel at the origin.
Code:
                |
              * * *
          *     |     *  P
        *       |       o (Rcosθ, Rsinθ)
       *        |  R  / :*
                |   /   :
      *         | / θ   : *
  - - * - - - - * - - - + * - - 
      *         |         *
                |
       *        |        *
        *       |       *
          *     |     *
              * * *
                |

The wheel has radius \(\displaystyle R.\)
The carriage is at point \(\displaystyle P.\)
. . \(\displaystyle \theta\) is the angle made with the horizontal.
Hence, \(\displaystyle P\) is at: \(\displaystyle \,(R\cos\theta,\,R\sin\theta)\)

Here's a close up near point P.
Code:
                  P
                  *
                  | \           *
                r | α \ r      *
       *          |     \    *
          *       + - - - *
                  *         Q

The height of the carriage is \(\displaystyle r.\)
The bottom of the carriage is at \(\displaystyle Q.\)
\(\displaystyle PQ\) makes angle \(\displaystyle \alpha\) with the vertical.

Relative to \(\displaystyle P\), the coordinates of \(\displaystyle Q\) are: \(\displaystyle \,(r\sin\alpha,\,r\cos\alpha)\)


The coordinates of \(\displaystyle Q\) are: \(\displaystyle \,(R\cos\theta\,+\,r\sin\alpha,\,R\sin\theta\,+\,r\cos\alpha)\)


Hope this helps . . .

 
ferris3.gif


I think I may be confused.
This is an image of what I am trying to do, though I couldn't find one that spins counter clockwise. At any rate, if this wheel were to spin counter clockwise, I am looking for the equation of the bottom middle of the carriage with respect to both the ground and the middle of the wheel, given the fact that the bottom of the carriage swings back and fourth as if someone inside it were rocking it (left and right in the picture). I added this last clause/sentence for clarifacation on my own part; having both equations will help me understand it better. Thank you for what you've done though! I will try to see if I can develop something with it, as it gave me a couple of ideas.
 
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