While I had the tub removed from the washer to make the brake repair I noticed a lot of dark material around the top of the tub, located between the basket (the perforated basket in which you place your clothes) and the external tub (the part of the washer that contains the water). Further investigation determined that this dark material consisted of dried on lint and dirt from the clothes. Evidently, as we wash our clothes a form of "bath tub ring" forms around the top of this tub. Over the years the material accumulates and forms a thin layer of "gunk" washed from our clothes; primarily lint, dirt, and soap. I suspect it is the soap that acts like a glue to hold all of this in place.
I just imagined how this layer of material would hold water and over time begin to harbor that musty smell which my wife complained about. Since the tub was out of the machine, it seemed like a rather convenient time to clean this gunk form the tub.
Before we go any further, let me take this time to add my disclaimers:
Disclaimer #1: The information posted is for informational use only. Use it at your own risk as the statements on this page may be completely wrong, or your washer may not be like mine, or your mechanical skills may be rusty, or...well, there are myriad reasons why you shouldn't attempt this repair or the information may not be valid. Caveat emptor!
Disclaimer #2: Unplug your washer before working on it. You don't want any electricity to be present when you open the cover on the washer and begin poking around on the inside.
The ideal thing to do would be to wash off the gunk with a high pressure water stream from a garden hose or perhaps a car wash. However, I had already installed the brake on the tub and didn't want to that wet. Also, it was rather cold outside--too cold to run a water hose comfortably--so I decided to scrape the material from the tub while remaining inside the comfort of the washroom.
The first thing I had to do was remove the rim from the top of the tub. This rim just clips on, so using a putty knife and screw driver to pry it off worked well. I placed the putty knife against the tub and then placed the screwdriver between the blade of the putty knife and each clip of the ring. The putty knife prevented the screw driver from puncturing the side of the tub. Sorry I didn't take any photographs of this process :-(
I used a putty knife and a pocket knife, but you might find something more effective. My two main concerns were (1) puncturing the tub, which would produce a leak during the next use of the washer, and (2) making the smooth plastic tub rough, which would just speed the accumulation of gunk to the tub in the future. So, very carefully, I scraped the material I could get to with my knives, pulling it from the tub placing it on a paper towel (i.e., wiping it from my knife) so it wouldn't drop down into the bottom where it might cause further havoc.
After I scraped all I could, I took a spray bottle of 1 part vinegar and 1 part water and sprayed the remaining gunk and then wiped it out the best I could. This vinegar/water solution seems to be a good all-around cleaner when I don't want to use soap. Vinegar is a good mild cleaner and works on most surfaces: windows, doors, cabinets, etc. Also, I figured that the chemistry of the vinegar would offset the chemical composition of the soap: vinegar is a mild acid and soap is a mild base (remember your pH scale from high school chemistry?). Hopefully the vinegar would break down the surfactants in the soap and clean the plastic surface of the tub.
One thing to remember about soap is this: it is designed to attract dirt. This is a physical/chemical process that works very well. Unfortunately, once an item is cleaned, it is rather difficult to remove all of the soap from the surface. Many times, a bit of the soap remains, especially if water was not used to rinse the surface. So, any remaining soap tends to attract dirt quicker than would normally occur with a "clean" soap-free surface, therefore it becomes dirty quicker than previously.
This phenomenon occurs when you have your carpets cleaned or you use any of the household spray-and-wipe cleaners, etc. That soap residue that is left behind attracts dirt. However, when using the vinegar, no soap is used so no soap residual remains to attract dirt.
Once the majority of the gunk was removed I gave the tub a good spraying of vinegar/water and re-installed it back in the washer. I suspect that spraying the inside of the tub with the vinegar solution after using the washing machine will help prevent the build up of the gunk too. Another option is to add some vinegar to the washer and let it run through a wash cycle, but getting a 1:1 mixture of vinegar and water in your tub will require several gallons of vinegar, which is probably not cost effective and could damage the washer pump or other components.