Problem $3(45$ points$)$

A radial saw base with a mass of $36\mathrm{.}48$ kg is driven by a motor running at $200$cycle$\frac{s}{m}in,$ producing a $10-N$ force due to a small imbalance. The resulting vibration was detected after manufacturing, and the manufacturer wants to retrofit a vibration absorber to eliminate the table's oscillation.

a$)$ Design the absorber$,$ assuming the table legs provide an effective stiffness of $3200\frac{N}{m},$ and the absorber must fit inside the base with a maximum deflection of $0\mathrm{.}25$ cm $.$

b$)$ Calculate the bandwidth of operation such that the amplitude ratio $1$

c$)$ Add a damper to the absorber to increase the bandwidth of operation. Find the damping ratios which satisfy these requirements? What is the new bandwidth range for the same amplitude ratio? $.$