Fitting driven Harmonic Oscillator I've got some datapoints of a turning disc. It is supposed to obby the following differential equation:
$I\ddot{\theta}+\gamma\dot{\theta}+k\theta=\tau$,
So it should have the form of a driven harmonic oscillator.
Now my question is how do I get to know the angle at which it will be at equilibrium? I know $I,k$ and $\gamma$. Even numerically would be a good solution.
The data is a set of (t,y) points which clearly resemble the evolution of a driven, damped harmonic oscillator.
Thanks!
 A: By definition of the equilibrium point, its time-derivatives must vanish (the equilibrium point doesn't change with time), so $\ddot{\theta}_\mbox{eq} = \dot{\theta}_\mbox{eq} = 0$ and the differential equation then tells you that $\theta_\mbox{eq} = \tau/\kappa$.
Added after more comment exchanges:
Since you have $(t,\theta)$ points, you can estimate $(t,\dot{\theta)}$ and $(t,\ddot{\theta)}$ by using numerical methods to evaluate derivatives and then do a linear regression on the data to estimate $\tau$.
Beware of accumulated errors if you follow this path. I'd strongly recommend reading up on numerical methods. One excellent resource is this.
Alternatively, you can solve the differential equation analytically and fit the data to obtain both $\tau$ and $\theta_{\mbox{eq}}$. Now, this can be very simple or very complicated, depending on the nature of your driving force and on the values of the known parameters. I'll look only at the simplest case:


*

*Constant driving force


If $\tau$ is constant, then the general solution of the differential equation is
$$\theta(t) = \frac{\tau}{\kappa} + \mbox{general solution of }\{I\,\ddot{\theta} + \gamma\,\dot{\theta} + \kappa\,\theta = 0\}$$
Since the homogeneous equation is such that its solution $\theta(t)$ satisfies
$$\lim_{t\,\rightarrow\,\infty} \theta(t) = 0$$
then all you need to do to estimate $\theta_{\mbox{eq}}$ is to look at the last data point in chronological order (assuming you have data for a long enough length of time that your system has reached its equilibrium state). From that value of $\theta$ you can then estimate $\theta_{\mbox{eq}}$ and $\tau$:
$$\theta_{\mbox{eq}} = \mbox{last }\theta$$
$$\tau = \kappa\,\theta_{\mbox{eq}}$$
If your driving force is not constant, then you may want to look at this page for some details on how the solution depends on the parameters, in the case of an under-damped oscillator driven by a sinusoidal external force (that's the next simplest case). Be aware of the fact that your $\gamma$ is not the same as that page's $\gamma$. Theirs is yours divided by $2I$.
A: If the system is at rest and in equilibrium, then $\ddot\theta=0=\dot\theta$ so $\theta=\frac{\tau}{k}$
