Naomi -- I've been discussing the idea (of, somehow, producing gravity waves by our own means) in a bit more detail with a colleague. My hope was that maybe we'd be able to pump out some numbers by the middle of next week, in terms of what sort of scale we were looking at, ie "can you produce gravitational waves of sufficient size with something humans could feasibly make?" The question of feasibility here will be limited strictly to a mass scale.
It's a nice thought, certainly, to be able to produce and harness gravitational waves ourselves. What for I don't know, but if you can do it then I'm sure a decent use could be found -- one such use might be a very long range message transmitter, as these waves can evidently propagate for billions of light years without losing coherence. But whenever you have to basically blow up a very large star in order to produce something useful I think it's pretty reasonable to doubt that it's likely to happen.
Anyway, if you're still interested -- and if me and my friend get the chance -- then we might attach some numbers to this.
A quick taster, though, from one source I was looking at (see
http://arxiv.org/pdf/1209.0667v3.pdf ):
"It appears to be utterly impractical with current technology to detect manmade gravitational waves [my note: this means that production is not in principle impossible, but production at a level worth writing home about is the hard part].. imagine a dumbbell consisting of two 1-ton compact masses with their centers separated by 2 metres and spinning at 1 kHz... one obtains a [gravitational wave] amplitude of 10^-38 [metres]." For comparison, the waves LIGO detected had an amplitude some 100 million million million times stronger. Presumably a future detector might be able to push the threshold of detection down as far as maybe 10^-24, leaving a further 14 orders of magnitude gap to bridge. Good luck with that.
All the same, despite my claim that it is essentially entirely unfeasible to do anything other than observe gravitational waves produced naturally, it would be lovely if I was wrong. I don't think I'll live long enough to find out.