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Once the centre of mass moves behind the aerodynamic centre, the rocket will do everything in its power to flip over. Testing your rocket design suggests that fuel burn is likely to be causing this to happen at around 11,000-14,000m, depending on your ascent profile.
During the tests I found that it is possible to make your craft very stable, you just have to mount those fins at the very lowest point on the lower tank.
The reason this happens to Stayputnik and not the command pod is due to Stayputnik being 1/16th of the mass of the pod.
http://imgur.com/gallery/rYDrUm4/new
Are you running any mods? If not, the general rule on rockets is weight as for up as possible and all drag (i.e. fins) at the bottom.
I did several launches, going straight up, only hitting space bar to launch, it went to 350,000 meters. Doing a slight gravity turn, I nearly put it into a stable orbit.
I'll go back into the game and double check.
So confused now.
EDIT: Turns out the fins could have been lower. Moving as absolutely low as possible managed to make it stable enough to reach a suborbital trajectory, though it still felt like a near thing. Thanks for pointing that out. Still confused about this CoM thing, though. I thought the point was to get it as low as possible. I'll look up 'aerodynamic centre'.
Note that anything attached radially and not in a fairing induces quite a bit of drag (e.g. batteries). If you have them, putting stuff in a fairing makes everything much easier.
Also, if your TWR is high (like it is for your craft), then you'll want to throttle back or add more fuel so that you're not going to fast in the low atmosphere.
Finally, turn on fine control mode and try your best to keep the nose close to the prograde marker. If you start having trouble keeping control, reduce thrust a bit.
Get SETI. It's balanced around probes first. Parts, contracts, tree, everything.
Wouldn't you get a severe wobble as soon as you touch one of the pitch or yaw controls?
use a decoupler for a little nudge at start. you have no control, the rocket has to fly itself until the final stage. you might not even need the spin, I've been playing modded version where I needed it to keep it stable.
the launch decoupler force was set at 30% for this rocket.