A vibration reduction mechanism on a shaft could include a damper. The shaft is metal

with an elasticity that can be modeled as a spring $($ks$).$ The shaft end unit has a rotational

inertia $($Ip$).$ Inside is a viscous fluid around an inner rotational unit with an inertia $($Id$).$

This can be modeled with the following schematic:

Derive the equations of motion of the angular displacement of the shaft end unit $(p)$ and of

the inner unit $(d$ where the shaft angle $($the input$)$ is $$ s$.$