Shakes roughly during spin cycle

This assumes that you have a basic Kenmore / Whirlpool top load, direct drive, neutral drain (it doesn't spin until the pump removes most of the water) machine.

First, see the Sears parts site for your washer:
http://www.searspartsdirect.com/partsdirect/index.action

Enter your model number (###.######) and you'll see a list of major sub-components with diagrams and parts.

If you look at the section on "Machine Base Parts" you'll see a number of springs and vibration pads.

It's possible that one or more of the springs or pads has come loose or broken.

In order to get to them you have to remove the cabinet.

Here is a description of how to do it.

http://www.american-appliance.com/catalog/newsdesk_info.php?newsPath=16&newsdesk_id=131&osCsid=90fdc122f6335c7c64f46f2ea1b1c887


Also see the Whirlpool Service Manual for top load washers. It has a troubleshooting section that describes the reasons for excessive walking or vibration.

WHIRLPOOL INST AND MAINTENANCE INCLUDING HOW TRANSMISSION WORKS

https://secured.whirlpool.com/Service/SrvTechAdm.nsf/2cd44500d572193285256a45004fd9d6/f7e2147fa98b28cf85257122007475cb/$FILE/L-55.pdf


WHIRLPOOL TRANSMISSION DESCRIPTION AND SPRINGS

https://secured.whirlpool.com/Service/SrvTechAdm.nsf/2cd44500d572193285256a45004fd9d6/2550f737b3129b0c85256b0600504738/$FILE/787772%20%234.pdf


If it's not the springs or pads, you may consider that the "Agitator Cams" in the transmission are not raising the Agitator Gear high enough (about 1/16" ) when the washer goes into spin mode.

See the description of what the Cams do in the Service Manual.

We had the same problem, excessive vibration and noise during the spin cycle.

In order to solve that problem, :

Assuming that you know how to remove the cabinet, agitator (you don't need to remove the tub), pump, motor and transmission the agitator cam and follower use the same whirlpool part number 62580/01. This is a plastic set used to raise the agitate gear on the shaft.

Pull the transmission.

Remove the clutch (the spring holding the clutch to the spin pinion is a bear to replace.

Remove the 8 transmission screws carefully, the transmission is full (15 oz.) of 80-90 weight oil.

Remove the transmission cover.

Pull off the spin pinion.

Push down on the agitate spring and remove the spring clip. Note where is attaches to the main shaft.

The two plastic agitate cam parts pull right off. Don't lose the washer on top of the cams.

Replace the cams, washer, spring and clip, pinion, oil (assuming you drained the oil first), cover and clutch parts.

Line up the transmission so that the release cam on the brake drum will snap onto the clutch spring retaining clip and put all parts back in the washer.

The basic tools are screwdriver, small socket set with up to a 1/2" socket, vice grips or good pliers (to remove and replace clips)

It sounds a bit complicated but it really isn't too bad and (For a $15 part, saved buying a $170 transmission.)

Following is the Whirlpool parts list for their commercial washers although the consumer models use the same part numbers.

http://shared.whirlpoolcorp.com/assets/pdfs/literature/Repair%20Part%20List%20-%20W10114743.pdf