Time Machine will not backup an external hard disk in Ventura or Sonoma macOS

Per above, I erased the external hard disk (called HD2).

I finally turned off time machine (manual, no auto backups) and told it to exclude internal and external disks, and then erased my time machine disk, restarted, and then set up time machine, and ran the backup. Again, external hard disk did not backup.

There must be a preference corrupted or maybe a privilege of some sort for HD2?

By the way, same problem in Ventura before upgrading to Sonoma.

thanks

To access backed up versions of files, "Browse Time Machine Backups" is the command that I use in 'Time Machine'

I have been using Macs and Time Machine for a long time, and it's generally reliable to recover a file(s) that was changed or deleted. I have not had the problem of Time Machine just not backing up a volume that is on an external hard disk previously. I am sure that the external hard disk is not getting backed up while the macOS SSD is getting backed up normally by Time Machine.

I hope I'm being clear enough. thanks

Thanks for the response. Here is more info:

1) new Mac Studio set up in September and it backed up external HD (HD2) once and internal macOS SSD (ISSD), but only backed up ISSD thereafter. This occurred even after I erased HD2 and reformatted.

2) When I opened a folder in ISSD and Browse Time Machine Backups, I can scroll through all the backups since September, but only see the first backup for HD2 from September. After erasing HD2, there are no backups in Time Machine.

3) HD2 is a 2 TB hard drive which is less than half full

4) The Time Machine hard disk is 16 TB

5) The 2 external hard disks are in an OWC Thunderbay that has 4 slots. Time Machine has worked with the Thunderbay on another Mac as I recall.

By the way, I didn't realize until today that the HD2 backups were not being done

I suppose I could erase the Time Machine external hard disk and set up a new Time Machine backup - maybe that would work?

Problem appears to be the OWC Thunderbay since Time Machine worked fine with a 2 TB hard disk connected directly to the Mac Studio. Surprising since Time Machine storage disk is in the Thunderbay 4 (OWCTB3IVKIT0GB)

Thanks for the responses

I get sort of like the picture above but only "Today (Now) shows with no actual backups

The picture above is accessed per above by selecting "Browse Time Machine Backups"

thanks