Modal Dynamics Approximations of Cantilevers Under Partial Follower Load

Presented herein is a modified Galerkin discretization procedure for determining the qualitative dynamic behavior of elastic cantilevers with internal damping under partial follower step loading at their tips. For this strong nonlinear nonconservative system, the scheme proposed makes use of basic functions that are a product of nonlinear corrections of approximate linear shape functions. These corrected modes are computed in a way that all the nonlinear nonhomogeneous boundary conditions of the actual problem are satisfied throughout the motion. Numerical results obtained using a two-mode approach are found to be in very good qualitative agreement with the finite element results presented in the literature, not only in the vicinity of the critical states, but also in remote unstable domains. The effect of variation of initial conditions is also investigated and the advantages of the proposed procedure compared with conventional ones are discussed. Further research is required for establishing its capabilities and the range of its applicability for a broader class of nonconservative dynamic problems.