For the SDOF system shown below, spring $k_2$ is attached to the original system of $m,c,k$ and

displacement $A$ is prescribed.

$($a$)$ Construct the expressions for kinetic energy, strain energy and dissipation function.

$($b$)$ Use the Lagrange equation approach to construct the equation of motion for $y.$ Identify the

applied force term due to prescribed $A$ in the equation of motion.

$($c$)$ For $m=60kg,k_2=2400\frac{N}{m},A=A_{0}sint,$ resonance occurs when $=25ra\frac{d}{s}.$ Estimate

$k$ and the natural frequency of the original system without $k_2.$

Note: The damping is small and can be neglected for part $($c$).$