In all probability, you should be able to board the flight. I use the expression because this is not a matter of rules, but depends on various factors that cannot be predetermined, such as the punctuality of the flight that lands you into Frankfurt, the rush on the two flights, and the number of people doing the transition from this to that.
I had a similar experience once while flying Lufthansa from Bangalore to San Francisco. I had a little more time than you--90 minutes instead of 55, but guess what, my Bangalore-Frankfurt flight was late by around an hour. 30 minutes would have been impossible to make the connect but the airlines staff knew that, and they boarded all passengers of the incoming flight that had to connect to the next flight to San Francisco in a separate coach, took them to a special window where they got their passports stamped, and then back onto the coach and the plane for the connecting flight. I think the connecting flight was even delayed by a few minutes as 30 minutes would not have been sufficient to transfer 50+ passengers. It was smooth for passengers, though a little tiring of flying 20 straight hours.
One of my two check in bags didn't make it though. I got only one bag in SFO, and the other came to my hotel the next evening.
