Extort money from, as in They had quite a racket, shaking down merchants for so-called protection. [Slang; second half of 1800s]
Make a thorough search of, as in They shook down all the passengers, looking for drugs. [Slang early 1900s]
Subject a new vehicle or machine to a tryout, as in We'll shake down the new model next week.
Become acclimated or accustomed, to a new place, job, or the like, as in Is this your first job? You'll soon shake down. [Mid-1800s]