If I wanted to measure the oscillation rate of a barrel (at any point or interval) as it is discharging, what equipment would I use?

Your tongue????

Tonque - I love it

Anyway - multiple accelerometers mounted along the barrel at each axis with a multi channel high frequency recorder and associated amplifiers - I'm an aerospace test engineer so give me a break. Accelerometers mounted on top of the barrel would measure up and down movement at that location (velocity and duaration - sign wave) and those along the side would measure left and right movement. Plug the data into a computer with the right s/w and it will generate grapahic model. pricey solution. Glass bed the end of the barrel - its cheaper