How to find force parameters if you know the solution?

[jonjacson]

Hello folks.

Normally if you have two variables and you want to know their relationship, you can plot them to get an idea of the equation that might relate them. If you see a straight line you can use the least squares method to find the best equation. You can use this method for polynomials higher than one.

But my question is of different nature.

Assume we know the Drag Force can be represented by a funcion that depends upon the velocity, and we assume that its highest degree is 2. So the force is something like this:

F_d = A + B * v + C * v²

Where A, B and C are constants to be calculated using real data and v is the speed.

My question is, if I am given a real path, How could I calculate the best values for A, B and C?

I don't know how to do it because these parameters are put into a differential equation, Newton second law, to calculate the path and there are more forces in it, like gravity for example. But I don't know to work in reverse order, I don't know how to get the parameters starting from the path. So if I knew A,B and C I can calculate the path, but not the opposite.

I hope you understand what I mean. I guess there must be a mathematical technique for this since I am sure I am not the first one solving these type of problems.

 

[BvU]

HI,

to work in reverse order

Suppose you have a path, in the form of a set of observations $(t_i, x_i)$ and suppose you have initial guesses for $A, B, C$ -- call them $A_0, B_0, C_0$. Furthermore, you have a model:$$m\ddot x = A + B\dot x + C\dot x^2.$$

So if I knew A,B and C I can calculate the path

means you can calculate the model predictions for the $x_i$ for $A_0, B_0, C_0$. Call these $x_{i,m}$.
Next thing to do is minimize the differences between the observed $x_i$ and the predicted $x_{i,m}$ by varying $A, B, C$. If all measurements are equally accurate, we usually minimize $\displaystyle \sum_{i=1}^N (x_i - x_{i, m})^2$

The method is called parameter estimation, in my example based on least squares minimization.

I left undetermined how this is done: sometimes it can be done analytically, but often numerical methods are needed.

 

[jonjacson]

Thanks, I have just found books on "parameter estimation", I guess I can learn what I want there.

I really appreciate your answer.