Maybe it sounds stupid but is there anyone out here who did a big trip on an electric motorbike?
If so: which bike, how and where did you charge, is solar power an option?

I'm an Electrical engineer... And I have been looking at options. Right now I'm traveling RTW but I a couple years I might try out electric options.

Current bikes have a max rang of about 150 miles. So if you did this in a day, it wouldn't be limiting. Quick charging usually takes 2 hours, but regular charging is overnight. Unless you added more batteries, the range might be a problem in some countries. I would think traveling around Europe on an E bike would be great though. I think offroad will be a limitation for any electric bike. Sure there are offroad electric bikes, but the range goes way way down.

I've been keeping an eye on the Brammo Empulse Brammo 2014 Empulse & Empulse R
I like this one because it still has a 6 speed transmission. I like the idea of shifting. I think a single speed electric bike would limit fun.

Zero Motorcycles also makes some decent bikes. A friend of mine has the Zero FX and he loves it. He puts it on the back of his overland rig and charges it while he drives from camping spot to camping spot. 2014 Electric Motorcycle Model Line || ZERO MOTORCYCLES

2013 was the first time an electric motorcycle won overall at the pikes peak hill climb. The Lightning electric motorcycle beat every single gas powered bike. When you think about it, it makes sense. Amazing torque at all ranges and you don't loose ANY power with elevation. Its a sprint not an endurance race so that also benefits the electric bike.
Lightning's Electric Superbike takes 2013 Pikes Peak International Hill Climb

Charging with solar panels is feasible for someone who is using the bike as a commuter. The solar array would be to big to carry with you though. Maybe if you went very small like an electric bicycle, but then you are charging during day and riding when? 1 day ride, 1 day charge? Ride in morning and evening, charge in middle of day?

I like the idea of plugging it in at a hotel, hostel, or house where you sleep. Give them a 2 or 3 $ for the power and continue in the morning. If I was doing it, I would even have a meter to show them how much power I took so I could compensate accordingly.

I think the recent announcement of the Harley Livewire project is huge. What - Harley is actually doing something innovative..? Apparently so.
From reading the Cycle World "first ride" article, it sounds like a decent motorcycle, held back only by very limited range. Then again, this is just a first prototype, so I expect to see a more reasonable riding range if it ever makes it into production.

I think I could get by with a 150 mile range if the batteries charged quickly. Ride for 2-3 hours in the morning, find a place to charge the batteries while you take a siesta, ride another 2-3 hours in the afternoon, charge overnight and repeat.

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Bruce Clarke - 2020 Yamaha XV250

Considering that electric power (as in the stuff you get out of a plugh on a wall/ceiling) is unreliable in some parts of the world (think asia) and non existent in others it may be a bit limiting for around the world travel. Fine for commuting, weekends. But anything adventurous .. not yet. Hard enough to get fuel in some parts.

looks like electric movement is big:
ZERO MOTORCYCLES – The Electric Motorcycle Company - Official Site

Zero range I think is 170 miles on charge.

I think the 170 miles is city riding, not highway. Electric bikes, unlike petrol, get better mileage in the city then on the open road.

This guy gave it a go when battery life was considerably shorter......Kanichi Fujiwara - Wikipedia, the free encyclopedia

"The scooter he used in the 2004–2008 circumnavigation, sponsored by Yamaha,[10] weighed 45 kilograms (99 lb), had a 30 km/h top speed, and an endurance of 20 km on a battery charge.[4] Even with six batteries giving a 100 km range,[4] his partner had to shuttle charged batteries to him in order to cross Australia's Nullarbor Plain.[6]"

He gave it a go, sponsored by Yamaha and had someone bringing him batteries on a bike that has a top speed that is barely speeding in a school zone

No thank you. I'd rather pic a push bike if I had a choice between the 2.

At this stage,no,it's not an option.But hopefully soon.
But I may not live long enough.

Cheers