I thought this was a trilogy from Michael Moorcock - but maybe not. I can't remember the name of the trilogy or any of the books (the one I read was a three-in-one volume, like The Stainless Steel Rat was).
In book 1, the wisest political and scientific minds on Earth forsee a complete annihilation of the human race on Earth, and build a super spaceship to carry a remnant to a far distant planet - and no hyperspace here... The people were to be kept in cryogenic stasis for a time.
I think the second involves some of the people waking up for some reason.
The third involves several generations later when the people on the spaceship have begun to worship the ship itself (and its computer) as supreme beings, and have no knowledge of the actual facts (that they are on a computer-controlled spaceship vessel). One of the people on the ship dares to "go behind the curtain" (and violates all kinds of religious rules) and finds that they are actually on a computer-controlled ship, and that something has gone wrong and they are about to crash. He saves the day somehow.
UPDATE: Could be by Ben Bova too. I can't remember for sure.