Instructions: Solve problem below. Derive your solution expressing one equation per line in a

single column. Clearly identify the step and solution.

A rigid bar OA of length $l=0\mathrm{.}2m$ and mass $m=5E-03kg$ is connected to a torsional spring

of stiffness $k_1=0\mathrm{.}075N\frac{m}{r}ad$ and two linear springs of stiffness $k_1=0\mathrm{.}220\frac{N}{m},$ at distance

$a=0\mathrm{.}09m,$ and $k_2=0\mathrm{.}220\frac{N}{m},$ at the end, as shown in Fig. $1.$ Using $$ as the generalized

coordinate:

$1\mathrm{.}1.$ Derive the equation of motion of the uniform rigid bar. Hint: Since $$ is measured from

the static equilibrium position, the gravity force need not be considered.

$1\mathrm{.}2.$ Calculate the natural frequency of vibration.

$1\mathrm{.}3.$ Assuming the displacement of the system is represented by the real part of the rotating

vector, for the initial conditions ${}_0=-0\mathrm{.}1$rad and ${}_0^{}=0\mathrm{.}12ra\frac{d}{s}$ :

$1\mathrm{.}3\mathrm{.}1.$ Calculate the amplitude of free vibration and phase angle.

$1\mathrm{.}3\mathrm{.}2.$ Find an expression for the displacement $(),$ velocity $({}^{}),$ and acceleration $({}^{})$

$1\mathrm{.}3\mathrm{.}3.$ Calculate the maximum velocity and acceleration.