Time machine ejects disk and fails to recognize USB drive on iMac

The "Disk Not Ejected Properly" message is usually linked to either a cable or drive enclosure issue or a power issue, i.e. the drive and other connected devices may be drawing more power than the port can provide.

• Try connecting the backup drive to another port.

• If the cable can be removed from the drive enclosure, swap it with another high quality cable.

• Connect the drive directly to the Mac, not through a hub or dock.

• If you have other devices connected to the computer, try running with those disconnected.

• If you are connecting through a hub or dock, use one that has its own ac power adapter.

You can check the System Information... (option-click the Apple  menu > System Information...) to check the power requirements of the USB ports on your Mac. Below is a snapshot of one drive and connector on my Mac. I'm pretty sure I can't provide power for any additional devices on that USB port.

Full Computer Sleep Mode can cause external drive ejections and other issues.

If the drive(s) still eject after working thru D.I. Johnson's cable, connection and power requirement advise.

Then test the following System Settings:

Turn On "Prevent automatic sleeping when display is off"

Set "Turn display off when inactive" to 5 or 10 minutes

Leave the iMac On and only allow the display(s) to turn Off

Do not manually Sleep from the Apple Drop Down menu

Shut Down the iMac when away for a day or more

This is somewhere in the power settings that if an external disc is idle, it will disconnect. My 2 external drives do that at least once per day, often more. Nothing to worry about, and don't disconnect it. As soon as your computer needs to use the drive, it knows it is still actually connected.

My Time Machine drive behaves exactly as you describe. Once a week when it is supposed to do another backup, it automatically connects and does the backup. Later that day it will say it was not disconnected properly. I ignore the message, and the following week, it knows it is still connected without me unplugging or plugging anything in. My second external is for my photography. It sometimes displays the message even when I have the editing program open but only if I am not working on a photo. Move the mouse, and it reconnects.