Padded helmets have not been seriously tried, to the best of my knowledge. Insulate the head from trauma to a greater extent and concussions would decrease. Otherwise, why do cars have airbags?
You might reduce the number of concussions by changing the nature of the impact.
Look I majored in engineering at Carolina, and I'm going to try and explain this (though things like this can be tricky when you get into the details). I hope I don't bore anyone, and if someone knows better I hope they correct me.
First in a concussion event, the brain has to contact the skull. It has tissue around it inside the skull, but that can give way. If you take one of those Mexican shakers with the bean inside it, you get the same effect. If you stop the shaker suddenly the bean inside continues on the path it's momentum carries it until it contacts the wall of the enclosure it is in.
Now no matter what you do, any possible helmet design or padding has to deal with certain physical laws. One is that there is so much momentum in a moving object (like a sprinting linebacker). In the case of a dead on collision that stops the linebacker in his tracks when he hits something, that is going to be transferred to what he hits. Now the nature of the collision (elastic like him running into a bunch of pillows, or inelastic like a brick wall) also has some bearing on what happens. But let's consider the collision to be helmet to helmet in this case, which is reasonably close to an inelastic collision.
Okay, the linebackers job is done. Your helmet head combo now says something like "Okay the laws of physics say I now have momentum in this direction. Darn it, until something stops it I have to go in this direction at the speed my mass and the imparted momentum say. It's the law, no way around it."
Ignoring the fact that we have a number of components, this is the point where your brain is like the bean in the Mexican shaker. Your brain is at rest until something acts on it to move it ("An object at rest, tends to stay at rest."). In this case the helmet moves. It transfers force through the padding to your skull. That moves. Then moving it encounters your brain (ignoring the tissue padding for simplicity's sake). There is a collision between your skull and your brain.
Okay, no one is happy about this, but at least everyone is moving in the same direction. Then something stops the helmet. Maybe your neck or the ground. The helmet stops. The brain is still moving. It contacts your skull again.
Now that sounds kind of nutty. It has happened to everyone if they experience anything that moves their head rapidly enough (or stops their head rapidly enough). Part of life, you can't get around it. But it doesn't happen anywhere nearly as frequently as in an activity like football.
Now the padding can make a difference. I'm not going to ... no way around it really. It can basically act as a dampener of sorts by changing the nature of the collision (at least from our brain's standpoint). It can also reduce acceleration by making things "take longer."
I'm sorry, this isn't hard. But I'd have to make some drawings. Plus while writing about this, I also think about how what I've written isn't QUITE the way it actually works, as well as the fact that my model is too simplistic.
But people a lot more knowledgeable than me about this subject have undoubtedly already worked it out.
Suffice it to say, that you can help a little with padding, but no matter what you can't get around F=MA and the laws of motion.
