Dynamic stability of a tapered cantilever beam on an elastic foundation subjected to a follower force

The equation of motion in matrix form of a tapered cantilever Euler beam subjected to a follower force at the free end is formulated based on the Lagrangian approach and the assumed mode method. The beam is resting on a Winkler-type elastic foundation. The effects of the presence of viscous damping in the foundation, which makes the foundation viscoelastic, and the presence of internal damping in the beam on the critical flutter loads are examined separately to evaluate their relative importance. As expected, internal damping of the beam tends to drastically reduce the critical flutter loads for a beam of uniform cross-section. For tapered beams, the effects are however dependent on the taper ratio of the beam as well as the modulus of the elastic foundation. The critical flutter loads of both tapered beams and beams of uniform cross-section are found to be unaffected by the presence of viscous damping in the elastic foundation. The effect of varying the modulus of the elastic foundation on the critical flutter loads is also discussed in the paper.