Time Machine does not recognize my backup disk

I'm sorry to hear you're experiencing issues with Time Machine not recognising your backup disk. Below are some troubleshooting steps you can take to resolve the issue:

Preliminary Checks

1. Check the Disk Format: Time Machine requires the backup disk to be formatted as `Mac OS Extended (Journaled)` or `APFS`. You can check the format using Disk Utility.

2. Disk Space: Make sure your external hard drive has enough space to accommodate your backup.

3. Check Cables and Connections: Ensure that the external hard drive is properly connected via a good-quality USB or Thunderbolt cable.

4. Update macOS: Sometimes, updating your operating system can fix compatibility issues.

Troubleshooting Steps

1. Restart the iMac: If you haven't already, try restarting your iMac. This often solves many hardware and software issues.

2. Time Machine Preferences:

- Open Time Machine preferences from System Preferences.

- Turn Time Machine off.

- Disconnect the external hard drive.

- Reconnect it and then turn Time Machine back on.

3. Manual Disk Selection:

- Open Time Machine preferences.

- Click on "Select Disk" to manually select the backup disk.

4. Check Disk for Errors:

- Open Disk Utility (you can find it by using Spotlight with `Command + Space` and then typing "Disk Utility").

- Select your external hard drive from the list on the left and then click on "First Aid."

5. Console Logs:

- Open the Console app from Utilities.

- In the search bar, type “backupd” (which is the process name for Time Machine) to look for any related error messages.

6. File System Check:

- Unmount the disk from Disk Utility.

- Run `diskutil verifyDisk [diskname]` in Terminal.

- If issues are found, you may need to repair the disk using `diskutil repairDisk [diskname]`.

7. Reformat Disk: As a last resort, you may need to erase and reformat the disk. **Please note that this will delete all data on the disk.**

8. Contact Apple Support: If none of the above solutions work, you may need to seek professional help.

Perhaps time has passed to consider replacement of storage drive

for purposes of Time Machine backups on there, dedicated to Mac.

Back up your Mac with Time Machine - Apple Support

Tech 101: How to Schedule Time Machine Backups - MacSales.com

Start, pause, or resume a Time Machine backup on Mac - Apple Support

Back up your files with Time Machine on Mac - Apple Support

Time Machine (macOS) - Wikipedia

How to Speed Up Your Time Machine Backups - How-To Geek

Almost any 2TB or larger drive may be adequate; of better quality

brands, and also try and use other backups; maybe some clones?

My macs all have at least spare too, in case they need restored.

Types of disks you can use with Time Machine on Mac

Restore items backed up with Time Machine on Mac

Choose a backup disk and set encryption options on Mac

Connect a new backup disk to your Mac

Time Machine troubleshooting on Mac