Out of boredom* I derived some simplified equations to describe the dynamics of a pneumatic gun. These equations are extremely simplified, but they should provide some insight into the system that my numerical models can't. This system had one non-linear equation that could be rewritten as the following after some manipulation.(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\ddot{z} (C_2 z - C_5 t^2 + x_0) = C_4 + C_1 \tau (1 - e^{-t/\tau})[/itex]

[itex]z = \int_0^t \int_0^t P(t) dt dt[/itex]

z is pressure in the barrel integrated twice.

I am certain the t^2 term can be dropped as the projectile is launched in less than 30 ms for the slowest launchers I have made. Edit: Okay, maybe not as the x_0 term is actually reasonably comparable to the t^2 term.

A few different methods to obtain an approximate solution were attempted. I attempted to obtain a perturbed solution and nearly had success. Unfortunately, one term in each iteration of the series grew exponentially larger, indicating that this solution does not converge. I could provide the work for this if someone would like to check it. I have little experience with perturbation theory aside from what Farlow's PDE book says.

Linearization of the non-linear term with an average does not seem to preserve important physical characteristics of the solution (i.e. what is pressure grows asymptotically when it should grow to a peak, decrease, and then oscillate) and the resulting solution is fairly messy.

Replacing z in the non-linear term with the Taylor expansion about t=0 yields another t^2 term, which as I've noted, isn't helpful (or too accurate).

I looked at book of exact solutions for ODEs (about 6200 I believe!) without any luck in even finding a particular solution. I've also tried to add small terms to fit the equation to a form from the book, but due to some restrictions, this wouldn't work.

I've tried a number of guess solutions, guess change of variables, and even tried integrating the equation directly (after some appropriate change of variables) to see if I could drop any terms integration by parts leaves for the non-linear part (the answer to that is obviously no). I tried a number of other things I can't remember now too.

I figure I could guess a form of the second derivative in the non-linear equation and substitute it in to get a linear equation with variable coefficients, but that would make things a little messier. I'm going to plug in an approximate solution for z that I found unacceptable earlier to see if this would result in something reasonable. Maybe a few iterations of this would return a good solution?

Any ideas on how to get a reasonably correct approximate solution?

* This is what happens when I unexpectedly get a week off due to snow.

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# Stuck on second order non-linear ODE

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