Find the a$)$ governing equation and b$)$ the natural frequency, ${}_n,$ of the following system.

Show all your work.

Perform the following steps $($Do steps $1-10$ symbolically, don't plug in numbers until step $11)$

Coordinates and Kinetic Energies of Masses

Identify all of the masses. Are they rigidly connected?

Label a coordinate for each mass in the direction of its motion. Find a relationship between

each of the coordinates with the other coordinates. Assume small deflections.

Find the total kinetic energy of the system in the coordinates of each mass. $($Neglect the rotary

inertia of the box$)$

Find the kinetic energy of a rigid bar with equivalent kinetic energy as the total of the box and

bar of the system.

Find the equivalent polar mass moment of inertia, $J_{eq}.$

Potential Energy of Springs

Find the potential energy of the three springs in the system.

Write an expression for the potential energy of the equivalent torsional system in the same

coordinate chosen in $4)$

Find the equivalent torsional spring constant by using the expression $7)$

Governing Equation

Using Newtonian Mechanics $($Kinetic Equilibrium$)$ find the governing equation using the

equivalent inertial term found in $5)$ and the equivalent spring constant found in $8)$

Write the natural frequency as the root of the ratio the inertia term, $J_{eq},$ to the stiffness term

$k_{\text{t}\text{e}\text{q}\text{.}}.$

Plug in the numbers to get the numerical values for the governing equation and the natural

frequency. $k_1=k_2=1\mathrm{.}0k\frac{N}{m},k_t=1\mathrm{.}0kN-m,L=1\mathrm{.}2m,m_{\text{b}\text{a}\text{r}}=20kg,m_{\text{b}\text{o}\text{x}}=30kg.$