USB 3.0 external hard drive won't stay connected

Hi,

I'm not sure whether this is a problem with my Mac (late 2009 21.5 inch iMac, OSX 10.10, 3.06 GHz Intel Core 2 Due, 8 GB RAM) or my exernal hard drive, but I am having issues keeping my hard drive connected (WB digital passport, 500GB).

My hard drive will turn on when connected via USB 3.0, and will work for about 1 minute before disconnecting. In this time I am able to read and write, until the message pops up saying it was abruptly disconnected.

When the hard drive disconnects, I can still hear it spinning, and the indicator LED is still on. If I physically disconnect it, I have to leave it for 30 minutes or so before reconnecting, otherwise my Mac won't recognise it, which indicates to me that is might be something with the hard drive itself.

I have some large files that I'd rather not lose, and so bought a new hard drive but, of course, the old one doesn't stay on long enough to transfer them across.

Any thoughts, or ways to possibly salvage my files?

Cheers