Many problems that occur with any electronics happen due to power problems. LaCie drives are powered by external power supplies. The external power supply is a small, black box that contains a transformer. This changes the AC power coming from a power outlet, into the small amount of DC energy required by a drive. They appear somewhat like this:
A power supply failure can have these symptoms:
Try a different power supply unit, from LaCie and of the same specifications, if the drive has similar symptoms.