I am studying mass-spring-damper systems with coulombs friction. There are multiple discussions on simulating such systems using numerical methods and the problems that arise due the discontinuous excitation but I wanted to know if an analytical solution exists. To be mathematically clear about the problem, I am trying to analytically solve the following.
m*(d2x/dt2) + c*(dx/dt) + k*x = F*sign(dx/dt)
where the sign function is defined as:
sign(var) = 0 if var = 0
sign(var) = 1 if var > 0
sign(var) = -1 if var < 0
Note: I am aware of treating such systems as piece-wise linear nonlinear systems but I want to know whether a general solution exists that is capable of solving the problem without breaking it to a number of mini-problems.
If no general solution/analytic solution/closed for solution exists, we recourse to some sort of approximation. Related to your query, I suggest you follow : http://personal.psu.edu/axb62/PSPDF/discontODE00.pdf
https://math.stackexchange.com/questions/2228719/are-analytic-solutions-of-differential-equations-realistic
Amir Shahhosseini:
Asking or seaching for "general analytic solutions", "exact solutions", etc is mostly a worthless (and often futile) task. If someone told the solution was a MeijerG function with such and such parameter sets (it isn't), what have you learned about the behaviour of the solution? Almost nothing! Only that the solution already has a name.
You may give a name, say AS(t), to the solution of your problem also. And say that this is the general analytic solution you search for. Now you just have to explore its properties. The function depends on many parameters, of course. So you better call it
AS(t; m, c, k, F; x0, v0, t0)
You should immediately be able to derive the property
AS(t; λm, λc, λk, λF; x0, v0, t0) = AS(t; m, c, k, F; x0, v0, t0) when λ ≠ 0,
and probably also that
AS(t; m, c, k, F; λx0, λv0, t0) = λ AS(t; m, c, k, F; x0, v0, t0) when λ > 0,
and further some similar trivial consequence of time translation invariance of your problem. Next you may want to know the t-values for which the time derivative of AS is zero, and the value of AS at these points. Compare this with finding the t-values at which the time derivative of a Bessel function Jν(t) is zero, and the corresponding values of Jν(t). What do you think is simplest? How much would it help you to know that you are dealing with a Bessel function (i.e., the general analytic solution of some problem)? Certainly more than nothing, because many others may have encountered the same problem before.
But isn't it much more satisfying to work with an entirely new function, and be the first to discover its properties??? To be the leading world expert on AS-functions!
In usually, the variational theory is applied for existence, uniqueness and solving them. Also, it is simple which one surveys these class problem in the their several regions that derivative of its solution is only in sign, by at means, they have to be non positive or non negative sign.
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