Influence of an elastic end support on the stability of a nonuniform cantilever subjected to dissipative and nonconservative forces

Abstract

The influence of an elastic end support on the stability of a damped, linearly tapered cantilever of rectangular cross section subjected to a follower-end-load is studied. The equation of motion is formulated within the Euler-Bernoulli theory for the case of a Kelvin model viscoelastic beam. The effect of external damping is also included in the partial differential equation of motion. The associated adjoint boundary value problem is derived and appropriate adjoint variational principle is introduced. This variational principle is used as the basis for determining approximately the values of the critical load of the system as it depends upon the taper parameters and the stiffness of the elastic end support. It is found that for a given damped, tapered beam the introduction of an elastic support may destabilize the system in certain eases. Furthermore, the critical value of the stiffness of elastic support at which the instability mechanism changes from flutter to divergence or vice versa is decreased as taper increases. For a given support, an increase in taper also decreases the value of the critical load of the system. These reducing effects are more pronounced with increase in thickness taper. The familiar destabilization phenomenon due to small internal damping is observed from the numerical results, but sufficiently large internal damping may have a stabilizing or destabilizing effect depending on the stiffness of the elastic support. © 1980.