Hi Scott,
Nice name! ;-)
The surest way to eliminate "whip" is to shorten your mandrel and turn only one barrel at a time. A lot of penturners do this. I don't. I like to be able to see the whole pen on the mandrel so I can visualize how it's going to turn out. So if you insist on using a long mandrel and turning both barrels at the same time, start by buying good mandrels from a reputable source. I usually get my mandrels from Berea, but lately I picked up one of the adjustable mandrels systems from Craft Supplies, and I like it real well.
The mandrel shaft might be bent a bit, and you can try to straighten it. Put your mandrel on the lathe and bring up the tailstock, but don't load any blanks. Turn on your lathe at a slower speed, and just barely touch the tip of your thumb on the spinning shaft. You should be able to tell if the shaft is bent. If it is bent, this is how I try to straighten it: With the lathe off, bring your tool rest up as clost to the mandrel as you can. Turn the spindle by hand and see if you can notice where it is bent in comparison to the tool rest. Another good way to see is by laying the tip of a pencil over your tool rest so it just slightly touches the shaft, and then turn by hand. Once you determine which way it is bent, you can gently push it back into line. Remember gently.
I hope this helps!
Scott.
