Filtering methods of the complex oscilliations

If I have a system of a springs, not one, but for example 3 degree of freedom system of the springs connected in some with each other. I can make a system of differential equations for but it is impossible to solve it in a general way. The question is, are there any papers or methods for filtering such a complex oscilliations, in order to get rid of the oscilliations and get a real signal as much as possible? For example if I connect 3 springs in some way, and push them to start the vibrations, or put some weight on them, and then take the vibrations from each spring, are there any filtering methods 开发者_JAVA技巧to make it easy to determine the weight (in case if some mass is put above) of each mass? I am interested in filtering complex spring like systems.

Three springs, six degrees of freedom? This is a trivial solution using finite element methods and numerical integration. It's a system of six coupled ODEs. You can apply any form of numerical integration, such as 5th order Runge-Kutta.

I'd recommend doing an eigenvalue analysis of the system first to find out something about its frequency characteristics and normal modes. I'd also do an FFT of the dynamic forces you apply to the system. You don't mention any damping, so if you happen to excite your system at a natural frequency that's close to a resonance you might have some interesting behavior.

If the dynamic equation has this general form (sorry, I don't have LaTeX here to make it look nice):

Ma + Kx = F

where M is the mass matrix (diagonal), a is the acceleration (2nd derivative of displacements w.r.t. time), K is the stiffness matrix, and F is the forcing function.

If you're saying you know the response, you'll have to pre-multiply by the transpose of the response function and try to solve for M. It's diagonal, so you have a shot at it.

Are you connecting the springs in such a way that the behavior of the system is approximately linear? (e.g. at least as close to linear as are musical instrument springs/strings?) Is this behavior consistant over time? (e.g. the springs don't melt or break.) If so, LTI (linear time invariant) systems theory might be applicable. Given enough measurements versus the numbers of degrees of freedom in the LTI system, one might be able to estimate a pole-zero plot of the system response, and go from there. Or something like a linear predictor might be useful.

Actually it is possible to solve the resulting system of differential equations as long as you know the masses, etc.

The standard approach is to use a Laplace Transform. In particular you start with a set of linear differential equations. Add variables until you have a set of first order linear differential equations. (So if you have y'' in your equation, you'd add the equation z = y' and replace y'' with z'.) Rewrite this in the form:

v' = Av + w

where v is a vector of variable, A is a matrix, and w is a scalar vector. (An example of something that winds up in w is gravity.)

Now apply a Laplace transform to get

s L(v) - v(0) = AL(v) + s w

Solve it to get

L(v) = inv(A - I s)(s w + v(0))

where inv inverts a matrix and I is the identity matrix. Apply the inverse Laplace transform (if you read up on Laplace transforms you can find tables of inverse of common types of functions - getting a complete list of the functions you actually encounter shouldn't be that hard), and you have your solution. (Be warned, these computations quickly get very complex.)

Now you have the ability to take a particular setup and solve for the future behavior. You also have the ability to (if you do things really carefully) figure out how the model responds to a small perturbation in parameters. But your problem is that you don't know the parameters to use. However you do have the ability to measure the positions in the system at repeated times.

If you put this together, what you can do is this. Measure your position at a number of points. First estimate all of the initial values of the parameters, and then all of the values a second later. You can adjust your parameters (using Newton's method) to come close enough to the values a second later. Take the measurements from 5 seconds later and use that initial estimate as your starting point to refine your calculations for what is happening 5 seconds later. Repeat with longer intervals to get all of your answers.

Writing and debugging this should take you some time. :-) I would strongly recommend investigating how much of this Mathematica knows how to do for you already...