I may have more experience with doing throttle body synchronization than anyone else on this forum.
Why? Because when I got my bike it vibrated excessively
in certain rpm ranges as you say your does. And when I did get a vacuum balance tool and started playing with it, I found that I could effect where (rpm range) the buzziness occurs at. And then I started experimenting with a bunch of different ways to get it perfect. But not just by adjusting the silly air screws, as shown in the Factory Service procedure. That procedure is pretty much a waste of time. All it does is balance the vacuum at idle, with an increasingly diminished effect as the throttle is opened.
The
Really Defintely Completely Unauthorized TBS procedure that Howie linked you to previously, is what I came up with that actually makes a difference. Even on bikes where the owner was previously satisfied with the amount of vibration, a noticeable improvement can be made. People that say a TBS doesn't do anything are not talking about this RDCUA TBS procedure, they are talking about adjusting the air screws, and I agree with them on that.
At NERDS tech day this year we did the RDCUA TBS on 5 or 6 bikes, of mixed generations. In every case the riders came back and told me that the vibrations were reduced in the cruising rpm ranges. How much of this was placebo effect is always uncertain, and how much actual difference there was would depend on how out of balance the TBS was to begin with. Most of them were pretty significantly out. But I can tell you that when I was first developing the procedure I was being hyper-critical of it, as I didn't want to tell people to go through something that would negatively effect their bikes. The improvement in vibration was very real and completely tangible.
Will the vibration go away in all rpm ranges? No!!
What ends up happening is that you will get the balance nearly perfect in one rpm range. When you are riding along you will notice a "sweet spot" where the vibration is completely gone. Ideally on a 2nd gen you want to shoot for that perfect smoothness near 4k rpm because that is such a useful rpm for cruising. In my last adjustment I somehow got the balance perfect at ~5k rpm with a little buzz left at ~4k and it starts to comes back a little at ~6k. This is a good compromise on a 1st gen with shorter gearing as 5k in top gear is 80-85 mph. There is actually more vibration on my bike right now at 70-75 mph than at 80-85 mph which is kind of cool (so long as the RD is working).
I would recommend that you read through that procedure closely and give it a try yourself. It is not all that difficult to do.
At various times I have also felt a need to re-torque the engine mounts, as previously mentioned. The way that I knew that was the problem was that the vibration would come and go at a steady rpm as you weighted and unweighted the suspension, riding over irregularities in the road. The engine is a stressed part of the frame system on and FJR, and with the alloy frame (which happens to get very hot BTW) there may be some minor dimensional changes that occur as it heats.
One thing that is worth adding to the torque procedure is to torque those engine bolts
after you have run the bike hard for a while. Get the frame good and hot, then bring it back to the garage and go through the 4 bolt sequence. This will work better to get the engine in that happy position in the frame at operating temps.
But... I could also be completely wrong... Right Skooter?