"Eureka - Pet Expert HEPA Bagless Upright Vacuum - Mystic Purple" is BRAND new and burning! I literally had soot on my hand and up my nose and mouth after using it just now. I've only had it for 24 hours. HELP!

Sounds like the beater brush is frozen and the drive belt is burning up.
Check your directions on changing the drive belt for the beater brush. When you remove the beater brush, clean as best you can. At each end of the beater brush you will find a plastic clip. These clips hold the beater brush in the vacuum. Add some WD40 between the brush and the clips. Turn the clips by hand and be sure they turn freely. Reassemble and at the same time, put on a new drive belt.
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