How to Solve These Nonlinear Equations?

Surely you don't expect people to pay $40 to try and answer your question....?

Why don't you contact the authors if you have questions...

• Andrés Kecskeméthy
• Annelie Weinberg

Please try this one:

https://www.researchgate.net/publication/227174074_An_Improved_Elasto-Kinematic_Model_of_the_Human_Forearm_for_Biofidelic_Medical_Diagnosis

(I tried to contact, but couldn't ...)

couldn't haven't yet tried hard enough

Please, try this link :

https://www.researchgate.net/publication/227174074_An_Improved_Elasto-Kinematic_Model_of_the_Human_Forearm_for_Biofidelic_Medical_Diagnosis

They seem to be talking about the stiffness, probably arrived at by testing or modeling...

The three positions are likely the range of movement, one extreme, midpoint, other extreme...

The Newton-Raphson method is an iterative approximation method for finding when an equation equals zero. I believe the author of the paper took a heuristic "best guess" for the initial values and then just iterated through the process until values converged enough that the error was tolerable. I expect the author's experience in the field of interest informed the author what the range of numbers will be likely to be close.

Good explanation; that confirms what I remembered from college back in the dark ages before computers. Iteration or successive approximation until the error is suitably small. You had to make a "sorta" decent initial guess in order for the method to converge.

You won't get much help here without letting us in on what the equations are. It is much easier to analyze a specific example rather than every possible set of non-linear equations.

If you have two equations, you only have solutions for two variables

If each of your two (dependent) variables, say x and y, depend on a third (independent) variable, say t, you can plug in a series of values into t and plot points (x,y) for each equation, and connect these points into a curve.

The points (x,y) where the curve for equation 1 crosses the curve for equation 2 are the solutions (x,y) of both equations.

Best I can do, completely in the dark...

I looked into the paper at work and the equation is a multivariate equation twelve pages deep into a medical paper. The paper is about the kinematics of the forearm in a human body but it is not really mechanical engineering. The non-linear aspect of the paper seems to be rooted in the array of trigonometric functions but since bones do not twist anywhere close to a full circle the range of initial conditions seemed limited and thus appropriate for a Newton-Raphson approximation. (I did not have time to perform a deep dive. I was at work. )

Make a first approximation.

Divide by 2.

Which ever value is closest, divide it by 2.

Continue with this process.

After 10 iterations you will be very close.