Grounding in most places was basically an afterthought after the electrical distribution system was installed, neutral should theoretically play the role of ground, but it's still a separate wire that runs back to the transformer, usually right beside the live/hot wire. It's possible that both could be damaged in such a way live could end up connected to neutral in your sockets, which, if the device used neutral as it's chassis ground (which, thankfully, they rarely do), would end badly. Having a separate wire that runs into ground straight outside your house means you're safe regardless of any faults that could arise on the line between you and the transformer (as even if you have a GFI installed, should the neutral line become damaged, it's essentially useless)
In some places they use an earth return system - basically very remote areas that would otherwise not be cost effective to run a full service power line to - they run just a single wire and use the ground as neutral/return. However the quality of the power is usually quite bad - so most places like this will have something akin to an online (or at the least, line interactive) UPS to get it back to an acceptable level.
As for the generator, it'll run fine without being grounded, but once again it's simply a safety precaution if for some reason the chassis becomes live. Luckily at 50/60Hz capactive coupling is pretty much a non issue, so the only real way it'd become live is through something like insulation breakdown. If the generator is used as a backup for powering your house (as in, you have a dedicated socket on the power panel to plug it in), chances are it'll already be connected to your grounding rod too.
In some places they use an earth return system - basically very remote areas that would otherwise not be cost effective to run a full service power line to - they run just a single wire and use the ground as neutral/return. However the quality of the power is usually quite bad - so most places like this will have something akin to an online (or at the least, line interactive) UPS to get it back to an acceptable level.
As for the generator, it'll run fine without being grounded, but once again it's simply a safety precaution if for some reason the chassis becomes live. Luckily at 50/60Hz capactive coupling is pretty much a non issue, so the only real way it'd become live is through something like insulation breakdown. If the generator is used as a backup for powering your house (as in, you have a dedicated socket on the power panel to plug it in), chances are it'll already be connected to your grounding rod too.
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