Question: ![enter image description here](https://i.stack.imgur.com/GHBz0.jpg)

So I'm a hobbiest who loves doing and learning about different subjects. Often I run into difficult problems that I normally can find solutions to however this one stumped me, I have reviewed many physics journals on the problem but either they are two complex or tackle the much easier version of 'viscous' friction.(p 2/54)
Essentially when I even begin a solution I yield to this:

Ma = sign(v) \* mg\*mu - kx

This would be simple if the friction didn't change direction. Does anyone have a suggestion on how to approach this problem? I'm not looking for someone to solve it I just want an idea on how to start it. Thank you

Answer: 
I would say Thibd's answer excellent. However, one should look at the cost (including fabrication) on Monel, Incoloy 825, any Inconel or Hastelloy or Titanium . Although not "steel" they are metals that would answer the intent of the question. Also in "deep" water, corrosion of even carbon steel is not a problem because of low oxygen; but I think testing would be necessary to determine if the the subject location falls into that situation. And, although not exotic, attaching an aluminum , magnesium or zinc anode to a carbon steel object would certainly work and be the cheapest option. One would just need to check literature for the size and type of anode.
You could chrome plate the steel and it would work fine. Steel rims that are plated with chrome last for a very long time on cars on roads that are heavily covered in salt during the winter.