Ok gravitational wave season, say it persists for 2 weeks, energy can be harvested for 2 weeks..
How much you'd get
Depend how far away you are, of course.. these waves have huge energies, but after 1 billion light years space time distance.. there wasn't much left for little Earth. Forbes calculated it,
"From a billion light-years away, two black holes of 36 and 29 solar masses merged, converting about three Suns worth of mass into pure energy. By the time those waves arrived at Earth, they had spread out so only 36 million J of energy impacted the entire planet: about as much energy as Manhattan receives from 0.7 seconds worth of sunshine. "
Now.. suppose such a gravitational wave persists for two weeks (you propose that, I'll handwaive), and the Forbes numbers are accurate, the wave that occurrered came from 1 billion light years away and it provided Manhattan (59.2 square km2 ) with 100% extra sunshine, for 0.7 seconds.
(disclaimer: below calculations are not accurate, more like "in the order of..")
What would be needed in terms of distance, to harvest that energy ? how near should your planet be, to get 100% extra sunshine, during two weeks, everywhere ? That would be 510,100,000 km² Earth's surface, not 59.. we'll need a factor 10 million times more energy to warm up the whole Earth. Also, you'll need to multiply the 0.7 seconds to get 2 weeks continuous energy so together, you're about 0.7/(24 * 3600 * 14)=5.78e-7, multiplied say you would need about 10e14 times more energy from the gravitational wave..
From an earlier topic I learned energy of gravitational waves is proportional to square distance so you need to take the square root of 10e14, that would be about 10 million. You should be 10 million times nearer.. the extra sunshine would happen if the two black holes would collapse at a distance of ca 100 light years, that is one billion light years (the original distance) divided by 10 million.
Concluding: at 100 light years distance, you can harvest 2x sunlight for 2 weeks everywhere
But again, your assumption about the two weeks spread MUST work.. if you don't have that spread, you'd be completely fried chicken when distance is only 100 light years and you'll get all energy at once, you'd have 10 million suns in the sky for 0.7 seconds.
To harvest it..
Night side of the planet would receive the extra energy too, it would be available 24/7
The Newton answer would be: you'd have to transfer very small mechanical displacements to larger ones. Tiny, but moves that allow huge, near-infinite reaction force. A very strong force. You could try to devise a gear system, to transfer that force to a rotator (generator) in some way.
The Einstein answer I don't know. It may be possible to harvest energy directly, from the distortion of space time itself, gravitational deviations.. Say you drop a large weight, will it bounce back up again, and how to derive energy from that.. and how to prevent your workers from bouncing up and down too?
Creating perpetual motion (newtonian) using the gravitational wave
With gravitational waves, spacetime will show a standing wave in a certain direction, like you have at the sea shore, blue is background, red is the wave added by the collision,
Gravitation will vary, in a certain pace. How to harvest this ? Let's say the wave will be in a certain direction, you'll have a band over your planet where the variation is perpendicular to the planet surface. Space time theory sais, you'll have a gravitational force proportional to the gradient of the spacetime surface. An observer will notice a variable gravitational force when spacetime has a hight gradient (goes steep).
Now suppose a large weight will be pushed upward, during a phase of small gravitation.. it would cost little energy, when the gravitational wave is in the downgoing position..
When this thing would fall back with enhanced gravity, it would bounce back with more force, getting up higher than before. If you repeat it a few times, in the same rhythm, your heavy object oscillates ever higher up. Perpetual motion. You could harvest the energy when it is touching the ground and bouncing back, or in the end, collecting all energy gained in the process.
The mechanism could also be used to launch objects into space.