Dynamic analysis of a tapered cantilever beam under a travelling mass

We study the vibration of a tapered cantilever (Euler–Bernoulli) beam carrying a moving mass. The tapering is assumed to be parabolic. Using the Galerkin method we find approximate solutions in an energy formulation that takes into account dynamic mass-beam coupling due to inertial, Coriolis and centrifugal effects. The approximate solutions are expanded in terms of the mode shapes of the free tapered beam, which can be obtained analytically. We then study the effect the tapering as well as the magnitude and velocity of the mass have on the tip deflections of the beam. We consider two different initial conditions, one where the mass starts moving from a statically deformed beam and one where the beam is initially triggered to vibrate. We find that tip deflections are more irregular for strongly tapered beams. Our results are of interest for barreled launch systems where tip deflections may adversely affect projectile motion.