1991 Nissan Sentra Starting problem

Electrical problem
1991 Nissan Sentra 4 cyl Front Wheel Drive Automatic 115K miles

Hello, hopefully you can help. I own a 1991 Nissan Sentra, with approximately 115K miles on it. The problem I'm having is intermittent. While turning the key to start, it cranks, then after about a second or two, it quits cranking and lets out a high pitched, loud winding sound. The lights come on, the door and seat belt alerts sound. Everything electrical works, except when it comes to actually trying to start. Sometimes, it just wont crank at all. I have replaced the ignition relay thinking that was the problem, but it continues. I've tried starting it in neutral, but I get the same results. The starter is a rebuilt unit, installed 9/2008. The battery was new, installed 10/2008. Could this be a relay, injector, or ignition switch problem? It sounds like the starter kind of "quits" trying to start after a second or two. It will not keep trying to turn the engine over.