$1.$ A simply supported countershaft $($E$=209$GPa, diameter$=25$mm$,$ density$=7800$ kg$/$m $3)$ with

two pulleys $($m$1=10$kg$,$ m$2=20$kg$)$ is shown in the Figure$1.$ A $2$ D$.$O$.$F$.$ lumped mass system is

used to model the system.

$($a$)$ Find the stiffness influence matrix and flexibility influence matrix.

$($b$)$ Determine the lowest critical speed using Raileigh$$s and Dunkerley$$s methods.

$($c$)$ Repeat part $($b$)$ by including the shaft weight Figure $1$