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Wolfram|Alpha solved a function that does not behave as expected?

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Dear Wolfram Community,

For my work I am using a mechanical system and have to find the relation between a linear displacement x and a certain angle alpha. To find x as a function of alpha is fairly easy and results in the following function:

Equation 1

Where H = 80mm, S = 205mm, a = 68mm, and z = 204.5mm are all system parameters. x is the linear displacement and alpha is the input angle.

I also need the inverse of this function (alpha = f(x)), but had troubles deriving this function by hand. Hence, I tried to have Wolfram Alpha find the inverse for me. This results in quite an extensive formula, which I'm ok with:

Equation 2

However, if I supply this function with a certain linear displacement, I do not get the expected (right?) answer. E.g. with an alpha of 40 degrees or 0.698 radians I get a linear displacement of 51mm (using the first equation), which seems to be correct. However, using 51mm as input for alpha = f(x) gives -3.018.

What am I doing wrong?

Thank you for your help. Matthijs

POSTED BY: Matthijs Damen
Answer
1 month ago

Might not be doing anything wrong but note that you have an inverse function that comes from a many-to-one function, hence loses other candidate inverse values. In particular multiples of 2*Pi might be added, and a negative of any solution is also a solution for the arccosine formula (whether it solves the original or is a "parasite" solution is another matter; this goes with the territory when inverting many-to-one functions).

POSTED BY: Daniel Lichtblau
Answer
1 month ago

Thank you for your reply, Daniel. I thought about that as well and tried adding and subtracting multiples of Pi. Also, I solved the first equation for several points within a range of alpha between 45 and -45 degrees (which is the actual range in the mechanical system). Using the inverse to find the angle again results in a range that is much smaller than 90 degrees total, which also seems strange to me.

POSTED BY: Matthijs Damen
Answer
27 days ago

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