# SDOF system

1. Feb 5, 2017

1. The problem statement, all variables and given/known data

m= 3kg, r=9cm, k = 21 N/m2, c= 63 N S/m
1. The smaller radius is half of the larger radius.
a. Find the equations of motion.
b. If the block is displaced 10 cm to the right and released from rest, find the angular displacement of the disk.
c. What are the natural frequency and damping ratio of the system in terms of m, c, and k?

2. Relevant equations
Equilibrium equations
$m \ddot x+ c \dot x + kx = 0$

3. The attempt at a solution
a.)

$F1 = (4k + k )x= 5kx$

Moment about the support:
$T (0.09/2) - 5kx (0.09) =0$
$T = 10 kx$

My positive x displacement is to the right. I'm a bit unsure at this point. Since T is expressed in terms similar to a spring force, I thought maybe I could treat it as such as such to find k equivalent.
$k_{equivalent} = k + 10k = 11k$
$3 \ddot x+ 63 \dot x + 11(21)x = 0$

b.
The conditions I see are:
It's from rest, so initially $\dot x =0$ also $x =0.1$
I don't know what variable I'm supposed to be looking for.
I don't know how to proceed. I think I need to find the displacement of the mass first, so that I can relate it to the angular displacement because of $r \theta$.