it sounds like a problem with the compters usb port. also it could be a problem with the cord itself, and lastly it could be the port on the iPod shuffle. i would recommend a teardown if you know what your doing. but more information would be nice. :)
its in essence a USB cord. they cost like 5 dollers. a teardown would most likely be a waste of time. im sorry for sounding rude. but its a cable with two different commections. take off the white plastic on the connectors and under them is the rest of the plug and its connected to wire cable, under the protective plastic of the cable is... copper or nickle wire. i recommend tape. or a new one.