Damped Driven Oscillator Dimensional Analysis So, I am studying the damped driven oscillator with the following drive force:
\begin{equation}
M\ddot{x}+\gamma \dot{x}+kx=F_0 \cos({\omega t})
\end{equation}
where $M$ is the mass, $\gamma$ is the damping force, $k$ is the spring constant and $\omega$ is constant. 
What I am trying to do is find a dimensionless expression of the equation above which can be done by applying the transformation:
\begin{align}
& x \to \xi=\frac{x}{x_c} \\
& t \to \tau=\frac{t}{t_c}
\end{align}
where $x_c, t_c$ are some "x, t characteristic". Now, I found that the expression of the oscillator in those variables would be:
\begin{equation}
\ddot{\xi}+\left( \frac{\gamma t_c}{M} \right) \dot{\xi}+\left( \frac{k t_c^2}{M} \right) \xi=\left( \frac{F_0}{M} \right)\frac{t_c^2}{x_c}\cos({\omega t})
\end{equation}
and by solving the initial equation I can acquire the analytical solution from which I could deduct that (for the case of $\Delta <0$):
\begin{equation}
t_c=\sqrt{\frac{M}{k-\frac{\gamma^2}{4M}}}
\end{equation}
which proves that everything is in the right place since it does match the period of the damped oscillator. 
My question is the following
What about $x_c$? I am not able to find an expression for it, and I can see that it is directly involved in the ODE. 
Thank you all!
 A: So, I think I figured it out. 
For the dimensionless expression, the constant coefficient $\frac{F_0 t_c^2}{Mx_c}$ of the drive force $\cos({\omega t})$, stands for the oscillation length. Taking that into account, along with the fact that we do demand $x_c$ to be dimensionless, one could think:
\begin{equation}
\frac{F_0 t_c^2}{Mx_c}=\lambda \Leftrightarrow x_c=\frac{F_0 t_c^2}{M \lambda}
\end{equation}
where $\lambda$ of course is a dimensionless constant.
Next, something that did not occure to me while posting this question, is that $t_c$ could be expressed either on terms of the system's own period like above:
\begin{equation}
t_c=\sqrt{\frac{M}{k-\frac{\gamma^2}{4M}}}
\end{equation}
or on terms of the frequency applied to the system by the drive force, therefore:
\begin{equation}
t_c=\frac{1}{\omega}
\end{equation}
One can use both, and I gave to myself a treat by using the second (and simpler) to reach the final result:
\begin{equation}
x_c=\frac{F_0}{\lambda M \omega^2}, \quad t_c=\frac{1}{\omega}
\end{equation}
