A Trying to establish a connection between the 2 rotating quadrants

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The discussion centers on finding a closed-form solution connecting two rotating quadrants through angles alpha and beta. The original poster attempted to use right-angle triangles but was unsuccessful, leading to the creation of a polynomial function of beta in terms of alpha. They proposed a relation between alpha and beta, specifically $$\sin \beta = \frac 1 {K_2} \left( 1 - \cos \alpha + \sin \alpha \right)$$, and sought confirmation of its validity. A participant confirmed the correctness of this equation and noted the importance of considering segments on the vertical axis. The conversation emphasizes the mathematical relationships and geometric properties involved in the problem.
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establish connection between the 2 rotating quadrants
Hello All,

I struggled to find a closed form solution connecting the 2 quadrants via the angles alpha and beta. I filled the schematic with right angle triangles hoping to find a connection but to no avail.
In the end, I decided to create a polynomial function of beta in terms of alpha as I know the coordinates of the vertices. Could someone confirm that a closed form solution is not possible? Is there any way I could use a custom analytic function of beta in terms of alpha? Thanks!

mechanism_v1.gif


Relationship_between_alpha_beta_v1.png


mechanism_v1.gifRelationship_between_alpha_beta_v1.png
 
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I find the following relation between ##\alpha## and ##\beta##: $$\sin \beta = \frac 1 {K_2} \left( 1 - \cos \alpha + \sin \alpha \right).$$ See if you can confirm this using the method sketched below.

1736366783251.png

In the figure, ##L## and ##L_2## are the lengths of the chords of the two quadrants: ##L = \sqrt 2 R## and ##L_2 = \sqrt 2 K_2 R##.

##\theta## and ##\phi## denote the angles that the two chords make with respect to the horizontal. You can show that ##\theta = \pi/4 - \alpha## and ##\phi = \pi/4 - \beta##.

Thus, ##\sin \theta = (\cos \alpha – \sin\alpha)/\sqrt 2## and ##\sin \phi = (\cos \beta – \sin\beta)/\sqrt 2##.

##A, B, C, D, ## and ##E## label various points. Note that $$AC +CE = AB + BD +DE $$ where

##AC = L \sin \theta##

##CE = R – K_2 R##

##AB = 2K_1R##

##BD = K_2R(1-\cos \beta)##

##DE = L_2 \sin \phi##

Use these relations along with ##K_1 + K_2 = 1##.
 
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Hallo TSny,

Indeed the equation you derived is correct and matches the angles calculated from the program/sketch. Many thanks for your input. I neglected to consider in futher detail the segments on the vertical axis.
Much appreciated and Thanks!
 
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