How to know if the vibration system requires one degree of freedom or two? and how to pick the right coordinate to describe the movement? I want to know a trick that helps me understand oscillatory systems and how to pick the correct general coordinates that describe the movement, I tried everything but I still can't get the solution right, Like in this image which coordinate is a general coordinate is it ${\theta}\; or\;{\varphi}$? or both? "the disc is rolling without sliding"

 A: 
There is no trick.
If you take the linkage AC out the disk can rotate $~\theta~$ and move towards  $~x~$ ,independent of the rotation $~\varphi~$ . The kinematic equation form the „linkage“ is
$$x=\frac l2\sin(\varphi)\approx \frac l2\varphi $$
The rolling condition is
$$x=R\,\theta$$
You obtained two constraint equations for the three degrees of freedom $~x,\theta,\varphi~$ , thus you have one generalized coordinate, you can choose x or $~\theta~$ or $~\varphi$ for this problem . In general the generalized coordinates must „covered „ the space of motion without mathematical singularity

with the kinetic energy
$$T=\frac{I_d}{2}\,\dot\theta^2+\frac{I_r}{2}\,\dot\varphi^2+\frac{M}{2}\dot x^2$$
and the potential energy
$$U=-\frac{k}{2}\left(\frac{l\,\varphi}{2}\right)^2-F_d\,\varphi$$
where $~I_d~$ the inertia of the disk, , $~I_r~$ is the inertia of the rod and $~F_d~$ is the damper force.
with $~\varphi~$ the generalized coordinate you obtain the equation of motion
$$\ddot\varphi+\frac{R^2\,k\,l^2}{I_d\,l^2+I_r\,4\,R^2}\,\varphi+
\frac{4\,R^2\,k\,l^2}{I_d\,l^2+I_r\,4\,R^2}\,\dot \varphi=0$$
