Hi book66,
There are various physical ways to reduce your impact noise transmission problem to your neighbours downstairs, which vary as to the effort and expense involved, and effectiveness. As the problem is of some severity and longstanding, and is affecting your health and well-being, then unless you are able and willing to move (and leave the problem for someone else to deal with) you need to take some remedial action beyond feng-shui measures recommended by Sean.
So far as your neighbours are concerned, there is literally nothing THEY can do about the noise, beyond moving themselves: and if they were able to, I imagine they might already have done so. Also, I wouldn't necessarily assume that by banging the ceiling, your neighbours are "enjoying" it: banging may be their only effective outlet for anger and frustration, and feeling trapped. Or, if worse, the banging may also be revenge. And as sound is a "carrier" of emotional energies, it works both ways: because conversely, even though you might believe the problem is "beyond your control" this is seldom true. And knowing this, your neighbours are most likely to believe that you could do something about it, but don't care: that any noise you make is deliberate. When feelings like this have been running for years, each side tends to become bitter and entrenched in their views.
However with this form of party floor construction, very often, "upstairs" people have literally no idea how much impact noise they're making: because they cannot hear it from underneath, where every impact is magnified like striking a drum. Also, impact noise (low frequency) is transmitted via structural walls, so your neighbours will not only be receiving it via their ceiling.
So first, as Sean suggests, be aware of how heavily you and your family might be treading. Whether or not you are on speaking terms with your neighbours, and unless you have done this already, I'd suggest you tell them you want to resolve the problem and ask them if you could go downstairs, one by one, to listen to what your family sounds like walking around overhead. And try to get others in your family to do the same. It's a good idea also to get an impartial witness: perhaps a family friend sympathetic to both sides of the problem, to be down there with each of you. For this experiment, make no attempt to walk differently from how any of you would normally do.
In addition, asking if you could (each in turn) go downstairs to listen, would call your neighbours' bluff if the noise is not really as bad as they might be making out (though it seems unlikely). Of course, you cannot really tell from a brief experience, how it might be actually to LIVE underneath the noise: but you might be better able to imagine it, and going down to listen would pacify your neighbours to some extent, simply as a sign that you were taking their concerns seriously.
Noise on an upper level, that within a household might be quite acceptable -- especially with some degree of control or persuasion over other family members -- can be intolerable to another household below. And this situation always means that people receiving noise become very sensitised to it, but this tends to be used as an excuse by noise-makers, to believe there is no "real" noise (or it's beyond control, etc). So whatever you hear, needs to be multiplied by their sensitisation, to approximate their own experience of it.
All that said, the first rule in acoustic treatment is ALWAYS to reduce noise at source. And being aware of how you walk, and consciously walking quietly, can make a surprising difference. The impact noise(s) made by toddlers is often worst: their feet hit the floor like pistons, and they're unable at a very early age to control their movements. If you have a very young family, your moving to a ground floor apartment might be the best answer for all concerned.
The current fashion for bare wooden floors is the worst scenario for impact noise from overhead apartments. The very least you can and should do is to seal any joints in the floorboards (if butt-jointed), lay hardboard throughout, and cover the entire floor area in your apartment with good quality underlay and thick pile carpet or, in kitchens and bathrooms, with underlay and padded lino or pvc. But if you have butt-jointed floorboards and springy joists, it might not be enough. If you own the apartment you live in and are able and willing to spend some money on a better physical "cure", you can reduce the reverberating drumskin effect that timber floor construction usually has, by acoustically deadening it. This means lifting your floorboards and laying ashes or sand ("pugging") between the joists on your neighbours' ceiling, before replacing the floorboards and sealing the joints. But you need to watch out for loadings and ceiling fixings: get a structural engineer to look at the place, lift a floorboard and give you advice on a safe loading and depth for sand or ashes, before going ahead.
The best acoustic answer is to construct a "floating floor" on top of your existing floor, though this is also likely to be the most expensive. As it raises the floor level by several inches, it would also involve reducing the height of your doors and re-hanging them. A "floating" floor is basically like a raft. Traditionally it's constructed by laying battens on top of an insulating acoustic quilt on top of your existing floor. On top of the battens is another quilt, and on top of that, battens at right angles to the first. On top of these battens is laid your new floor -- with hardboard, carpets etc on top of that. Check out "acoustic floating floor" on the web: I just did so and there are now a number of patented versions.
Whatever you choose to do -- in addition to the feng-shui actions recommended by Sean -- I hope it solves the problem.
