Homework H$\mathrm{.}6.$F

Given: Block A$,$ having a mass of $m,$ is supported by two springs and two dashpots, as shown below. Let $x$ represent the motion of block A measured positively upward. When $x=0m,$ the springs are unstretched.

Find: For this problem:

$($a$)$ Draw a free body diagram of block A;

$($b$)$ Derive an equation of motion for the system for the system in terms of the coordinate $x,$ its time derivatives, and, at most, the following parameters: $g,m,c,$ and $k$; and

$($c$)$ Determine numerical values for the undamped natural frequency ${}_n,$ the damping ratio $,$ and the damped natural frequency ${}_d.$

VERTICAL PLANE

Use the following parameters in your analysis: $m=32kg,k=1800\frac{N}{m},$ and $c=120k\frac{g}{s}.$