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Finding an equation for the curve created by two pivoting arms.
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Jul30-11, 01:07 AM
Hi, I have a question I am hoping that someone here can help me out with. I have not been in school for a number of years and I can't remember a lot of my calculus. Now I have a real world problem that I need to solve and wish I could remember it.
There are two pivoting arms (AC and BE) that are connected to two vertices of a triangle CDE. (See attached Diagram 1) The triangle is pushed by a force parallel to the line formed between the ends of the arms A and B. As the triangle moves it is forced to turn 180 degrees because the vertices are attached to the arms. The goal is to have point D of the triangle travel parallel to the line AB, or as close to parallel as possible.
The size of triangle CDE cannot be changed. The length of AC and BE is an approximate size and so it can be changed though it needs to be close to the length given below. The length of AB could be anything.
EC = 12
DC = 7.21
ED = 7.21
ED an DC are rounded off. They were calculated using the Pythagorean Theorem. The distance from vertex D to the line EC is 4.
AC = BE = Approx 24 (+/- 4)
AB = ?
I have attempted to figure out this problem using trial and error on Google Sketchup, but it is very time consuming and I just can't get it close to exact. Diagrams 2 and 3 show the movement of the triangle as the arms pivot.
Can anyone help me out by figuring out a formula to calculate the distance AB? Or possibly an equation for the curve created by the movement of vertex D. If you could even steer me into the right direction that would be great.
Thanks in advance for any help with this problem.
If I have posted in the wrong area, please direct me to the correct area to post.
Jul30-11, 06:11 PM
Welcome to PF, yugemos!
The simplest method would be to cut the arms out of paper and simply measure up the result.
To calculate, first define a coordinate system.
Say, put A in the origin, take an x-axis to the left, and an y-axis downward (so you will have all positive numbers).
Then define a system of equations based on the unknown coordinates of each point, combined with the known lengths of the rods (using pythagoras).
It's quite a big system I'm afraid, but that will give you your result.
You should be able to find a numerical result reasonably easy, once you have the system of equations.
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