Context:

In this project, you will be asked to work on mechanical vibration in a system made of a circular disk attached at the end of a shaft, both shaft and disk inside a medium providing a damping proportional to the angular velocity. The system can be modelled based on a mass$-$spring and damper system. This system can be useful as a first introduction to either rotating shaft in water $($propeller$),$ or as to model part of a rotating shaft in oil. Of course, the model describe here is merely the introduction.

Important concept to consider:

Moment of inertia: The moment of inertia can be understood as the inertia of the rotating body, or the physical phenomena that prevent the motion $($for example, the inertia in a linear motion is the mass of the object; for rotation, the geometry of the object is also important$).$ It will be represented by the letter I, and you will have to work with the moment of inertia about the central axis of the rotation.

Shear modulus: In the current application, it is the pressure needed to produce an elastic displacement along the plane perpendicular to the axis of rotation. We will use the letter G to represent it and it will be a constant.

Polar surface moment of inertia: This is a geometric value based on the cross section of the shaft. The $*$symbol used will be J$.$

The global motion of the object will be described using the angular position $.$ The general equation you will consider to start your study will be Newton's second law in angular motion, thus

${\displaystyle \underset{?}{\overset{?}{}}}$vec$(M)=I{}^{}$

Where vec$(M)$ represent the torque applied on the shaft.