There should be two U-shaped mounts at the bottom that the radiator rests in, with a third U-shaped mount on the bottom side of the top shield that holds it down. They all have rubber inserts, are you missing some of them? I don't think the shroud really does anything to hold the radiator in place, it just fits over it.
This diagram is from the '61 parts manual but it should be the same for '62.
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