Why won't my Western Digital (2TB) My Book for Mac backup to time Machine?

Question

This past fall (2012) I was notified by apple that the hard drive on my 2010 iMac might need to be replaced due to issues.  Before doing so I was prompted to back up all of my data with an external hard drive using Time Machine.  I purchased a Western Digital 2TB external hard drive and set it up easily.  The setup parameters were such that the thing should backup my iMac to Time Machine on the hour every hour.  This worked well and also allowed me to restore my user preferences for nearly everything when I got my computer back with its new hard drive.  This was around October.
Recently (mid-March), my WD My Book no longer backs up to Time Machine.  The computer recognizes the drive, allows me to access it, but will not backup to Time Machine.  I have no hypotheses as to why this has started happening, but it just did.  I have tried shutting Time Machine off and turning it back on to the particular external hard drive.  This has not worked.  I have tried to prompt a manual backup, but the computer gets stuck when preparing to back the drive up.  It never actually commences the backup.
I understand that just like anything else that things fail.  I also understand that sometimes things fail within the first few months of purchase (half a year-ish in my case).  I just cannot understand what might be causing it.  Any speculation?
Here are some screenshots associated with my current problems:

Answer 1

If you have more than one user account, these instructions must be carried out as an administrator.
Launch the Console application in any of the following ways:
☞ Enter the first few letters of its name into a Spotlight search. Select it in the results (it should be at the top.)
☞ In the Finder, select Go ▹ Utilities from the menu bar, or press the key combination shift-command-U. The application is in the folder that opens.
☞ Open LaunchPad. Click Utilities, then Console in the icon grid.
Make sure the title of the Console window is All Messages. If it isn't, select All Messages from the SYSTEM LOG QUERIES menu on the left. If you don't see that menu, select
View ▹ Show Log List
from the menu bar.
Enter the word "Starting" (without the quotes) in the String Matching text field. You should now see log messages with the words "Starting * backup," where * represents any of the words "automatic," "manual," or "standard." Note the timestamp of the last such message. Clear the text field and scroll back in the log to that time. Select the messages timestamped from then until the end of the backup, or the end of the log if that's not clear. Copy them (command-C) to the Clipboard. Paste (command-V) into a reply to this message.
If there are runs of repeated messages, post only one example of each. Don't post many repetitions of the same message.
When posting a log extract, be selective. Don't post more than is requested.
Please do not indiscriminately dump thousands of lines from the log into this discussion.
Some personal information, such as the names of your files, may be included — anonymize before posting.