Mechanics Question on Vibrations

[serfinos]

Homework Statement

[URL=http://img240.imageshack.us/my.php?image=imgso5.jpg][PLAIN]http://img240.imageshack.us/img240/790/imgso5.th.jpg[/URL][/PLAIN]

A hoist mechanism is modeled as shown in the figure. The drum is driven through a gearbox, the mass and flexibility of which is modeled as shown. The load required to be lifted and the cable stifness are also given.
Choose the appropriate co-ordinates and write down the equations of motion.
Put these equations in a matrix form and using values of
m=100 kg
k=500 kN/m
r=0.5 m
I(drum)= mk2 where k(radius of gyration)=1.4r
obtain all the natural frequencies and corresponding mode shapes.

Homework Equations

There are not relevant equations , i have made an attempt to write the equations of motion but i think i am wrong. Could anyone suggest a way on starting it ?

The Attempt at a Solution

ΣF= ma => k1(x-2rθ) - κ2 (x+2rθ) = Μx this is for the Force
ΣΜ=Ι α => -κ1(x-2rθ) ? = Ι θ

If anyone knows how to derive the right equations of motion it will be really helpful.
It will be one equation for the Force and one for the Moment , taking the right co-ordinates as well. And after it will be solved with the method of matrices.

Thanks

 

[MechanicalMan]

Try using Lagrange's equations to solve for the equations of motion. It's a lot simpler to find the equations of motion with Lagrange's equations (providing you don't leave out any of the terms).

After you find them using Lagrange's equations, you can put it into a matrix, find the characteristic equation, and find the natural frequencies accordingly.

 

[serfinos]

Firstly thanks for replying , this is what i get :
2 equations for motion and i am struggling to get the one for the moment.
mx1=-kx1-2krθ
mx2=-2kx2 - 2krθ

do you know if i am on the right way ?