Problem$3.(25$p$)$:

In the shown system. A revolute joint connects the link and the dise at $B.$ Both the dise and the link are homogenous.

Note that:

$m_1=m_2=m_1$

$m_3=(\frac{l}{8})m$

$r_2=(\frac{\text{I}}{4})L$

where $m=3kg,k=400\frac{N}{m}$ and $L=400mm$

a$)(4$p$)$ Prove that the system is single degree of freedom.

b$)(9$ p $)$ Derive the equations of motion for the system.

c$)(12$p$)$ Select a set of state space variables and find the state space equations $($in matrix form$)$ for the system. Consider the displacement $x$ of mass $\frac{m}{?}$ as the outputs, and the force $u(t)$ as the input in the same mass $m.$ Show both the symbolic and numeric versions.

Solution