|If it's possible to open the case on the external drive, then perhaps try opening the case and removing the drive. Then re-insert/re-install it back into the case; ensuring that it is "securely" in its connector slot. Then (without closing the case at this stage) see if it will operate correctly with whichever computers you have to hand. If it does.. then replace the cover(s)//reseal the case...; and check again.|
If it still doesn't work - it is likely the drive (as Jeffiner suggests). However it "may" also be the connector on the motherboard of the drive case itself. There are some usb drive cases that have very poor record with the usb connector itself. Some of the usb sockets are poorly soldered on; some very fragile and fail almost as soon as a usb cable is inserted into it.
Either way (dodgy/faulty usb connector or duff drive) it would be a case of sending it back to seller for replacement if still under warranty; as you do not want to be repairing dodgy usb connectors; and if the drive is duff - then again it's the seller's responsibility.
A very lo-o-o-o-n-n-g shot... It "may" also be the actual usb cable you are using? Have you tried connecting the drive with other usb cables - to eliminate that possibility?