I need to precisely control the motion of a damped, driven (nearly) harmonic oscillator:
$$ \ddot x(t) + \alpha\dot x(t) + \omega_0^2 x(t) \approx V(t) $$
I use the $\approx$ symbol because this is a real-world system, and I'm convinced this equation of motion is not sufficiently accurate for my needs. Help me determine a more accurate mathematical model!
The damping term $\alpha$ is positive, and small compared to the resonant frequency $\omega_0$ (approximately ten oscillations per decay half-time). I can choose the driving term $V(t)$ arbitrarily, and I can measure $x(t)$ fairly precisely.
In order to characterize my oscillator, I measured the impulse response $x(t)$ by setting $V(t)$ to (approximately) a delta function $\delta(t)$. The resulting behavior $x(t)$ fits fairly well to an exponentially decaying sinusoid, as expected:
$$ V(t) \approx\propto \delta(t) $$ $$ x(t) \approx\propto e^{- k t} \sin\,(\omega t ) $$
I use the $\approx\propto$ symbols to express that my input $V(t)$ isn't perfectly proportional to a delta function (it's a square pulse much shorter than the resonant period of the oscillator), and my output $x(t)$ isn't perfectly proportional to a damped sinusoid (but it looks a lot like one over a few tens of oscillations).
However, when I checked the impulse response for different impulse strengths, I got a surprise: my oscillator is nonlinear! The plot below shows impulse responses for several different impulse strengths, normalized by the impulse strength:
The first cycle of oscillations match fairly closely, but the decay time of my oscillator depends strongly on the impulse strength! For example, changing the impulse strength from $1$ unit to $7$ units increased the decay time by a factor of $\sim 2$.
Interestingly, the larger impulses seem to decay into the smaller impulses. The following plot shows the same data as the previous plot, except that the responses are not normalized. Instead, each response is shifted in time until it (approximately) lines up with the other responses:
So, my system looks a lot like a damped harmonic oscillator, but the decay rate of the oscillation seems to increase as oscillation amplitude decreases. How should I modify my equation of motion to account for this behavior? What other measurements would be helpful to guide my choice?
For example, it seems to me that my measurements are consistent with a damping that increases as velocity decreases: $$ \ddot x(t) + \alpha\dot x(t) + \beta\dot x(t)|\dot x(t)| + \omega_0^2 x(t) \approx V(t) $$
but my measurements might also be consistent with a damping that depends on position:
$$ \ddot x(t) + \alpha\dot x(t) F(x)+ \omega_0^2 x(t) \approx V(t) $$
What measurements might help distinguish between these models? I can't change the system resonant frequency or the damping, but I can set $V(t)$ to whatever I want, as long as it doesn't break my oscillator by making $x(t)$ go too large.
EDIT:
vonbrand asked an excellent question in the comments: What does it matter what the "right" model is?
I should be more clear about how accurate I need my model to be. I'm trying to achieve a particular output signal $x(t)$ from my oscillator. I wanted to use my measured impulse response to calculate what input signal $V(t)$ would give this output, as illustrated here:
I'd like my actual output to match my desired output within a few percent for a few tens of resonant periods. In the small-signal limit, this seemed to be working:
(ignore the banding artifact; that's what happens when someone opens the door during the measurement)
This is the right shape for $x(t)$, but the wrong amplitude; I want to operate at larger amplitude. When I scale $V(t)$ up to higher values, though, this is what happens:
This was actually the first hint that my oscillator was nonlinear, and lead to measuring impulse responses for different impulse strengths.