Consider a damped pendulum whose equation of motion is given in general by $$m\ddot{x}=-\mu\dot{x}-kx$$ where $\mu,k>0$
Rewrite this equation as
$$\ddot{x}+2\gamma\dot{x}+\omega^2x=0,$$
where $2\gamma = \frac{\mu}{m}$ and $\omega^2 = \frac{k}{m}$.
If $\gamma>\omega$, the roots of this equation are real and distinct. Define $\beta^2=\gamma^2- \omega^2$. Then the roots are $-\gamma \pm\beta$.
I have no issue with this, the solution is of the form $$x(t)=e^{-\gamma t}\left(Ae^{\beta t}+Be^{-\beta t}\right).$$
Given some initial conditions $$x(0)=x_0 \quad \text{and} \quad \dot{x}(0)=v_0 $$ Does the system return to rest in a finite time?
I have no issue here as well. I can rewrite the solution to read as $$x(t)=Ae^{(-\gamma+\beta)t}\left[1+\frac{B}{A}e^{- 2\beta t}\right]$$ where the next time, say $t_1$ such that $x(t_1)=0$ is given by $$t_1 = \frac{1}{2\beta}\ln\left(-\frac{A}{B}\right)$$
Here is where I struggle though. I can picture what happens if $$\frac{A}{B}<-1$$ I can choose some arbitrary number, say -2, so that $\ln(-(-2))=\ln(2)$ and thus $t_1$ is finite. In other words given some initial velocity the pendulum will move from its initial position and without oscillating return to its equilibrium position after time $t_1$.
However, now I am stuck.
If $\dfrac{A}{B}\to -\infty$ then $t_1 \to \infty$ I can make peace with. Am I right in saying that the system is so over damped that this pendulum almost comes to a complete stop and sloooooooooooooowly over an infinitely long period returns to equilibrium? If so, my brain can picture that, and I am happy.
But now the cases I can't make peace with.
If $A=-B$, i.e. $x(0)=0$, so we start at equilibrium, regardless of my initial velocity $\dot{x}(0)=v_0$, it is true that $t_1$ is always zero as $\ln(1)=0$. In my brain, I can picture a pendulum starting at equilibrium position, flying with some crazy initial velocity, and that equation is telling me, yeah.. The next time it's at $0$ is now instantaneously at $0$.
So what happened? It's impossible that the system is so overdamped that it's almost not damped at all and the pendulum flies infinitesimally fast to the equilibrium and then stops suddenly without shooting past. I kind of almost want to say it's negatively overdamped.. and that can't make sense.. can it?
Similarly, If $-1 <\frac{A}{B} - <0$, then $t_1$ is negative.
Similarly, If $\frac{A}{B} = 0$, or rather as $\frac{A}{B} \to -0$ then $t_1 \to -\infty$ is problematic.
Self-explanatory ... If $\frac{A}{B} > 0$ then we have a ln of a negative number which is also problematic.
So my question is really this. Are there physical relationships to the other 4 cases or is it one of those "the maths is possible but the physics is meaningless" scenarios. i.e. If $\frac{A}{B} > 0$ then that equation cannot physically ever be representing an over damped oscillator... or at least not in this universe. I am really hoping that's the case.
TL;DR - What is happening with over damped pendulums? Or does the equation of motion become meaningless under certain initial conditions?
PS: I am putting this under homework because my brain might be missing something obvious that someone might illuminate. My background is in applied mathematics and fluid mechanics. For the life of me I can't picture this mentally or find literature online that addresses these issues and unluckily I am curious to know things that bother me.