How can I solve fit inverse trigonometric function to a set of data points?

Hi all, I have to find the equation using magic formula to model a friction damper to be used in ADAMS software for dynamic simulation.

Y=D*sin[C*arctan{(1-E)*B*X+E*arctan(B*X)}]

I am trying to solve the Magic Formula (tire simulation) for the variables B, C, D and E in the above equation.I have the damper characteristic curve from which I extracted data points. Now I have to fit a curve equation that would fit the behavior it.

If it is possible to fit any other equation (polynomial or trigonometric), would be helpful. Can I fit a spline and extract the equation in this case?

[Star Strider]

How do B,C,D, and E in the equation relate to B,C,D, and E in the plot? Are they scalar parameters, (x,y) coordinate pairs, or something else?

[John D'Errico]

I was wondering that myself. There is no A in the formula as written, so I was guessing that they were not related to the parameters of the function. It is good to use distinct variable names so as not to confuse things.

As I point out in my answer, I think when someone cannot write a clear explanation of a problem, it is because they have not really understood the problem itself enough to be able to solve it.

[Star Strider]

This also came up a couple years ago from a different person (I Answered it), and Prathap brought it to my attention in a Comment to it today. The OP then didn’t reply to a similar request for clarification, so we’re left wondering.

I agree. If someone can write a clear explanation of a problem, the only glitches are likely problems understanding how to code it correctly. That reduces to learning the subtleties of MATLAB syntax, the real purpose of this forum.

[Prathap]

Thanks for your reply. I shall re frame the question with better clarity.

Answer 1

[John D'Errico]

No. There are no magic formulas. Nor is there any magic in mathematics at all, or in any field. You get results based on the effort you put into it.

You cannot fit a spline and extract a trig equation from it. No magic here.

You could build a spline model, but there is no simple formula one can extract from it. You can use a spline to interpolate the data of course. Perhaps that is what you need, but you have not been very explicit about what you need, so it is difficult to know even the form of the model. (Note that one can use tools from the curve-fitting toolbox, the stats toolbox, the optimization toolbox, etc., to fit nonlinear models. You can also use splines, such as pchip or spline to build an interpolant. But I have not seen a clear explanation of what you need, so it is difficult to know what you really need.)

You have also not been very explicit about even the variables in your model. One might presume X is displacement and Y is force, but that is just a guess. Note that The picture you draw is not a simple function of X, in that it shows a function that is not even single valued.

As important, the function you show IS single valued, thus for any X, there is a unique Y that will result. So it makes no sense in the context of the picture you drew. So either you misunderstand the problem, or you have miswritten the model to be solved, or something is out of order here. Until you do explain the problem to be solved more carefully, with clearly identified variables, etc., you can never get a useful answer.

Very often a big part of the problem in building any mathematical model is simply sitting down and writing out the problem clearly. What are the variables? What are your goals? What do you need? BE CLEAR! Then think about the mathematical form you need in terms of those variables. Write it down using mathematics. (Don't just dump a "magic" equation on the page and expect everyone to understand what your teacher asked you for.) Finally, in cases like this, look for a tool (or tools) that can solve the mathematical problem you have posed.