A helical compression spring is subjected to continuously varying external load. In the most compressed condition the force on the spring is $400$ N$.$ After an $8$ mm of release the spring minimum force on the spring drops a value of $260$ N$.$ Assuming that oil tempered wire is used with a mean coil radius of $13$ mm$,$ determine a suitable size, required active coil and total coil numbers and free length for the spring to function safely without any type of failure with a strength safety factor of at least $1\mathrm{.}20.$ After design, check if the spring you designed is fit or not if the frequency of external spring load is $5$ Hz$.$

Make logical assumptions "only if$"$ not enough information is given. Show all the calculations in details.

You should use Shigley$$s Mechanical Engineering Design book as the source.