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The following ODE describes a certain air spring system, which consists of a box with initial volume, and it has a cylinder on its side. By moving the piston in the cylinder we can change the pressure in the box. I would like to study what does happen when we push the piston into the box (making bigger pressure), then we release the piston, so I assume it starts to oscillate. The parameters:
$$A: cylinder's area\\ V_0: initial\ box\ volume\\ p_0: initial\ pressure\\ x(t): the\ position\ of~ the~ cylinder\\ m: mass\ of\ moving\ part$$
Then I made the following equations:
$$p(t)=\frac{V_0p_0}{V_0+Ax(t)}-p_0\\ F(t)=m\cdot a(t)=p(t)\cdot A\\ a(t)=x(t)''=\frac{A\ p(t)}{m}\\$$

From these I have got the following second order non-linear differential equation: $$x(t)''=\frac{V_0\ p_0}{m}\cdot \frac{1}{x(t)+\frac{V_0}{A}}-\frac{A\ p_0}{m}$$ with replaced constants: $$x(t)''=\frac{a}{x(t)+b}-c$$ Then I tried to solve this equation by replacing $$x(t)''=u(x)\cdot u(x)'$$ Finally I have got this equation which I do not understand how to solve: $$x(t)=\int_0^t\sqrt{2\ a\ log(b+x(t))-2\ c\ x}$$

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  • $\begingroup$ I don't think that the integrale can be expressed with elementary functions so there's nothong you can do. $\endgroup$ Commented Nov 6, 2022 at 12:59
  • $\begingroup$ Wait, where's the spring force in your equation? $\endgroup$ Commented Nov 7, 2022 at 18:46
  • $\begingroup$ Also, are you assuming adiabatic process? (E.g. the temperature doesn't change) $\endgroup$ Commented Nov 7, 2022 at 18:48

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You probably won't find find an analytical solution here, but you can linearize the problem in a certain regime. Given your equation $$x(t)''=\frac{p_0 A}{m}\Big(\frac{1}{1 + A x(t)/V_0}-1\Big),$$ you can see that when $|x(t)A| \ll V_0$, we can Taylor expand for $$x(t)'' + \Omega^2 x(t) \approx 0,$$ where $$\Omega^2 = \frac{p_0 A^2 }{m V_0},$$ so you have a simple harmonic oscillator. The condition $|x(t)A|\ll V_0$ means your oscillator is approximately linear with frequency $\Omega$ whenever the maximum volume displaced by the piston is small compared to the total volume of the chamber.

If you really want to incorporate the non-linear effects, you can carry the Taylor expansion to higher order. The quadratic contribution should make the frequency of the oscillator larger than $\Omega$, as you could calculate with perturbation theory.

A final comment - non-linear oscillations are well-studied. This one is probably discussed in some books, provided you set up the Newtonian problem correctly (but why wouldn't the compressibility of the gas enter the picture? What if you replaced the air with helium, shouldn't your frequency change?) Likely, the only progress is approximation and perturbation theory, as I've just begun to scratch the surface of by indicating the linear oscillator regime.

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