I am looking for some non-complicated second order differential equations to illustrate certain techniques for control engineering. It doesn’t matter if the differential equations are linear or non-linear; also, simplified assumptions are fine. I’d like to have a wider pool of examples.
I am aware of the standard example of the spring pendulum, and also the rolling motion of a plane simplified defined by: $$M(t) = k \ \delta_A(t)$$ and $$M(t) = I_{xx}\ \ddot \Phi(t)$$ with $M(t)$ being the moment, $\Phi(t)$ being the roll angle acceleration and $I_{xx}$ being the inertia torque.
Question: Can someone direct me to more realistic examples of this nature, or is there a reference or text containing a list of this sort somewhere?