The usual story, I am turning a Stainless blank and the first two inches go great but it goes into resonance and holding the center with my hand calms it down for a few more inches then all hell breaks loose. 360 rpm, 0.004 inches/rev feed, 0.015 depth of cut (0.030 on diameter). Right now I am turning it straight and could use a follower rest until I start to offset the tailstock and do the final taper cuts. Any clever fixes?

Thank you,
Charles

Are you using insert tool bits? Use a 1/64R tool bit. That may help and double the feed.

After EVERY pass, loosen the tailstock center and relieve the thermal expansion of the barrel. If the barrel gets hot let it cool back to room temp.

Charles change over to HSS tool bits and experiment with the shape and relief angles. HSS uses less cutting force then carbide, IMO. (No science in that opinion just experience) Use a small tip radius as already suggested, .010" max. Vary your depth of cut, feeds  and RPMs to find the optimal combination. And as the barrel diameter changes, so also the combination will. Barrel turning is an art form of experience. And a robust lathe! That little Grizzly can do the job with understanding of its limits.

For anything longer than about 20" I use a steady rest in the middle. Blend in the two turned sections with a lathe file. It works on tapered barrels as well as just roughing one down straight. This is on a South Bend 9" with HS tools. The 1/3 HP motor just doesn't work well with carbide tools

I'm old and slow and so is my Barrel Lathe,  9" South Bend.
Starting from a Unturned Blank I find they are all Barrel Shaped, larger in the middle and the centers from the Maker are no where's near in alignment with the bore.  So I square up the ends and put true centers in, a Job and a Half to do using a Center Rest.
First cuts may not be full diameter cuts as a Thin side Thick side cut will set up vibration.  Once the O.D. is trued up then I can do full cuts usually around .020 deep using HSS and running in Back Gear about 60 rpm with the Tool Bit a tad above Center, use a 6" Rule to tell.  Fairly Course Feed, the Lathe will let you know with a Paint Brush lightly resting on the barrel for cutting out.  Dead Center in the Tailstock.
Now some things to consider..
Live Centers are nice but it's another Moving Assembly.
Never get the Barrel Hot enough to cause the Cutting Oil to Smoke, though I like the smell!  As the Barrel heats up it gets longer and if it can't lengthen then it will Bow.
Two steady rest are nice as you can get close to one in front of the carriage and slip one behind the carriage then pull the front one.

Turning the Taper by Offsetting the Tailstock,  I'm assuming you're going to be using Centers on Both Ends.  Really look into using "Ball Centers" as they allow more Bearing.  Think about offsetting the Tailstock, the Center is still on line with the Bed but your Barrel is Not.

All of the Above is from the School of Hard Knocks!
Good luck, you've got a lot of knowledge from the other replies 👍🏻 David

Well, what a mess, but a 35 chunk of lead and negative raked old carbide, 0.020 depth of cut, 0.004 feed, 220 RPM. I wish I knew how to cut lead to shorten the chunk from 3.5" long to 2" long (it is 6" dia OD and bore of 2.5").

Charles

A good old fashioned hand saw will make short work of chopping soft lead into smaller pieces.